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q-Special functions, a tutorial

TOM H. KOORNWINDER

Abstract A tutorial introduction is given to q-special functions and to q-analogues of the

classical orthogonal polynomials, up to the level of Askey-Wilson polynomials.

0. Introduction

It is the purpose of this paper to give a tutorial introduction to q-hypergeometricfunctions and to orthogonal polynomials expressible in terms of such functions. An earlierversion of this paper was written for an intensive course on special functions aimed atDutch graduate students, it was elaborated during seminar lectures at Delft University ofTechnology, and later it was part of the lecture notes of my course on “Quantum groupsand q-special functions” at the European School of Group Theory 1993, Trento, Italy.

I now describe the various sections in some more detail. Section 1 gives an introductionto q-hypergeometric functions. The more elementary q-special functions like q-exponentialand q-binomial series are treated in a rather self-contained way, but for the higher q-hypergeometric functions some identities are given without proof. The reader is referred,for instance, to the encyclopedic treatise by Gasper & Rahman [15]. Hopefully, this sectionsucceeds to give the reader some feeling for the subject and some impression of generaltechniques and ideas.

Section 2 gives an overview of the classical orthogonal polynomials, where “classical”now means “up to the level of Askey-Wilson polynomials” [8]. The section starts with the“very classical” situation of Jacobi, Laguerre and Hermite polynomials and next discussesthe Askey tableau of classical orthogonal polynomials (still for q = 1). Then the example ofbig q-Jacobi polynomials is worked out in detail, as a demonstration how the main formulasin this area can be neatly derived. The section continues with the q-Hahn tableau andthen gives a self-contained introduction to the Askey-Wilson polynomials. Both sectionsconclude with some exercises.

Notation The notations N := {1, 2, . . .} and Z+ := {0, 1, 2, . . .} will be used.

Acknowledgement I thank Rene Swarttouw, Jasper Stokman and Doug Bowman for com-menting on an earlier version of this paper.

Korteweg-de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The

Netherlands; email: T.H.Koornwinder@uva.nl. This paper is a slightly corrected version (see in particular

section 2.5) of sections 3 and 4 in T. H. Koornwinder, Compact quantum groups and q-special functions, in

Representations of Lie groups and quantum groups, V. Baldoni & M. A. Picardello (eds.), Pitman Research

Notes in Mathematics Series 311, Longman Scientific & Technical, pp. 46–128, 1994.

–2–

1. Basic hypergeometric functions

A standard reference for this section is the book by Gasper and Rahman [15]. In particular,the present section is quite parallel to their introductory Chapter 1. See also the usefulcompendia of formulas in the Appendices to that book. The foreword to [15] by R. Askeygives a succinct historical introduction to the subject. For first reading on the subject Ican also recommend Andrews [2].

1.1. Preliminaries. We start with briefly recalling the definition of the general hyper-geometric series (see Erdelyi e.a. [13, Ch. 4] or Bailey [11, Ch. 2]). For a ∈ C the shifted

factorial or Pochhammer symbol is defined by (a)0 := 1 and

(a)k := a (a+ 1) . . . (a+ k − 1), k = 1, 2, . . . .

The general hypergeometric series is defined by

rFs

[a1, . . . , arb1, . . . , bs

; z

]:=

∞∑

k=0

(a1)k . . . (ar)k(b1)k . . . (bs)k k!

zk. (1.1)

Here r, s ∈ Z+ and the upper parameters a1, . . . , ar, the lower parameters b1, . . . , bs andthe argument z are in C. However, in order to avoid zeros in the denominator, we requirethat b1, . . . , bs /∈ {0,−1,−2, . . .}. If, for some i = 1, . . . , r, ai is a non-positive integer thenthe series (1.1) is terminating. Otherwise, we have an infinite power series with radius ofconvergence equal to 0, 1 or∞ according to whether r−s−1 > 0, = 0 or < 0, respectively.On the right hand side of (1.1) we have that

(k + 1)th term

kth term=

(k + a1) . . . (k + ar) z

(k + b1) . . . (k + bs) (k + 1)(1.2)

is rational in k. Conversely, any rational function in k can be written in the form of theright hand side of (1.2). Hence, any series

∑∞k=0 ck with c0 = 1 and ck+1/ck rational in k

is of the form of a hypergeometric series (1.1).The cases 0F0 and 1F0 are elementary: exponential resp. binomial series. The case

2F1 is the familiar Gaussian hypergeometric series, cf. [13, Ch. 2].We next give the definition of basic hypergeometric series. For a, q ∈ C define the

q-shifted factorial by (a; q)0 := 1 and

(a; q)k := (1− a)(1− aq) . . . (1− aqk−1), k = 1, 2, . . . .

For |q| < 1 put

(a; q)∞ :=∞∏

k=0

(1− aqk).

We also write

(a1, a2, . . . , ar; q)k := (a1; q)k (a2; q)k . . . (ar; q)k , k = 0, 1, 2, . . . or ∞.

–3–

Then a basic hypergeometric series or q-hypergeometric series is defined by

rφs

[a1, . . . , arb1, . . . , bs

; q, z

]= rφs(a1, . . . , ar; b1, . . . , bs; q, z)

:=∞∑

k=0

(a1, . . . , ar; q)k(b1, . . . , bs, q; q)k

((−1)k qk(k−1)/2

)1+s−rzk, r, s ∈ Z+. (1.3)

On the right hand side of (1.3) we have that

(k + 1)th term

kth term=

(1− a1qk) . . . (1− arq

k) (−qk)1+s−r z

(1− b1qk) . . . (1− bsqk) (1− qk+1)(1.4)

is rational in qk. Conversely, any rational function in qk can be written in the form of theright hand side of (1.4). Hence, any series

∑∞k=0 ck with c0 = 1 and ck+1/ck rational in qk

is of the form of a q-hypergeometric series (1.3). This characterization is one explanationwhy we allow q raised to a power quadratic in k in (1.3).

Because of the easily verified relation

(a; q−1)k = (−1)k ak q−k(k−1)/2 (a−1; q)k ,

any series (1.3) can be transformed into a series with base q−1. Hence, it is sufficient tostudy series (1.3) with |q| ≤ 1. The tricky case |q| = 1 has not yet been studied much andwill be ignored by us. Therefore we may assume that |q| < 1. In fact, for convenience wewill always assume that 0 < q < 1, unless it is otherwise stated.

In order to have a well-defined series (1.3), we require that

b1, . . . , bs 6= 1, q−1, q−2, . . . .

The series (1.3) will terminate iff, for some i = 1, . . . , r, we have ai ∈ {1, q−1, q−2, . . .}.

If ai = q−n (n = 0, 1, 2, . . .) then all terms in the series with k > n will vanish. In thenon-vanishing case, the convergence behaviour of (1.3) can be read off from (1.4) by useof the ratio test. We conclude:

convergence radius of (1.3) =

{∞ if r < s+ 1,1 if r = s+ 1,0 if r > s+ 1.

We can view the q-shifted factorial as a q-analogue of the shifted factorial by the limitformula

limq→1

(qa; q)k(1− q)k

= (a)k := a(a+ 1) . . . (a+ k − 1).

Hence rφs is a q-analogue of rFs by the formal (termwise) limit

limq↑1

rφs

[qa1 , . . . , qar

qb1 , . . . , qbs; q, (q − 1)1+s−rz

]= rFs

[a1, . . . , arb1, . . . , bs

; z

]. (1.5)

–4–

However, we get many other q-analogues of rFs by adding upper or lower parameters tothe left hand side of (1.5) which are equal to 0 or which depend on q in such a way thatthey tend to a limit 6= 1 as q ↑ 1. Note that the notation (1.3) has the drawback that somerescaling is necessary before we can take limits for q → 1. On the other hand, parameterscan be put equal to zero without problem in (1.3), which would not be the case if weworked with a1, . . . , ar, b1, . . . , bs as in (1.5). In general, q-hypergeometric series can bestudied in their own right, without much consideration for the q = 1 limit case. Thisphilosophy is often (but not always) reflected in the notation generally in use.

It is well known that the confluent 1F1 hypergeometric function can be obtained fromthe Gaussian 2F1 hypergeometric function by a limit process called confluence. A similarphenomenon occurs for q-series. Formally, by taking termwise limits we have

limar→∞

rφs

[a1, . . . , arb1, . . . , bs

; q,z

ar

]= r−1φs

[a1, . . . , ar−1

b1, . . . , bs; q, z

]. (1.6)

In particular, this explains the particular choice of the factor((−1)k qk(k−1)/2

)1+s−rin

(1.3). If r = s + 1 this factor is lacking and for r < s + 1 it is naturally obtained by theconfluence process. The proof of (1.6) is by the following lemma.

Lemma 1.1 Let the complex numbers ak, k ∈ Z+, satisfy the estimate |ak| ≤ R−k forsome R > 0. Let

F (b; q, z) :=∞∑

k=0

ak (b; q)k zk, |z| < R, b ∈ C, 0 < q < 1.

Then

limb→∞

F (b; q, z/b) =

∞∑

k=0

ak (−1)k qk(k−1)/2 zk, (1.7)

uniformly for z in compact subsets of C.

Proof For the kth term of the series on the left hand side of (1.7) we have

ck :=ak (b; q)k (z/b)k

=ak (b−1 − 1) (b−1 − q) . . . (b−1 − qk−1) zk.

Now let |b| > 1 and let N ∈ N be such that qN−1 > |b|−1 ≥ qN . If k ≤ N then

|ck| ≤ |ak| qk(k−1)/2 (2|z|)k.

If k > N then

|(b−1 − 1)(b−1 − q) . . . (b−1 − qk−1)| ≤ 2k qN(N−1)/2 |b|N−k

≤ 2k |b|(1−N)/2 |b|N−k

≤ 2k |b|−k/2,

so|ck| ≤ |ak| (2 |z| |b|

−1/2)k.

Now fix M > 0. Then, for each ε > 0 we can find K > 0 and B > 1 such that∞∑

k=K

|ck| < ε if |z| < M and |b| > B.

Combination with the termwise limit result completes the proof.

–5–

It can be observed quite often in literature on q-special functions that no rigorouslimit proofs are given, but only formal limit statements like (1.6) and (1.5). Sometimes,in heuristic reasonings, this is acceptable and productive for quickly finding new results.But in general, I would say that rigorous limit proofs have to be added.

Any terminating power seriesn∑

k=0

ck zk

can also be written as

znn∑

k=0

cn−k (1/z)k.

When we want to do this for a terminating q-hypergeometric series, we have to use that

(a; q)n−k =(a; q)n

(qn−1a; q−1)k= (a; q)n

(−1)k qk(k−1)/2 (a−1q1−n)k

(q1−na−1; q)k

and

(q−n; q)n−k

(q; q)n−k=

(q−n; q)n(q; q)n

(qn; q−1)k(q−1; q−1)k

= (−1)n q−n(n+1)/2 (q−n; q)k(q; q)k

q(n+1)k.

Thus we obtain, for n ∈ Z+,

s+1φs

[q−n, a1, . . . , asb1, . . . , bs

; q, z

]= (−1)n q−n(n+1)/2 (a1, . . . , as; q)n

(b1, . . . , bs; q)nzn

×s+1φs

[q−n, q−n+1b−1

1 , . . . , q−n+1b−1s

q−n+1a−11 , . . . , q−n+1a−1

s; q,

qn+1b1 . . . bsa1 . . . asz

]. (1.8)

Similar identities can be derived for other rφs and for cases where some of the parametersare 0.

Thus, any explicit evaluation of a terminating q-hypergeometric series immediatelyimplies a second one by inversion of the direction of summation in the series, while anyidentity between two terminating q-hypergeometric series implies three other identities.

1.2. The q-integral. Standard operations of classical analysis like differentiation andintegration do not fit very well with q-hypergeometric series and can be better replaced byq-derivative and q-integral.

The q-derivative Dqf of a function f on an open real interval is given by

(Dqf)(x) :=f(x)− f(qx)

(1− q)x, x 6= 0, (1.9)

and (Dqf)(0) := f ′(0) by continuity, provided f ′(0) exists. Note that limq↑1(Dqf)(x)= f ′(x) if f is differentiable. Note also that, analogous to d/dx (1− x)n = −n (1− x)n−1,we have

f(x) = (x; q)n =⇒ (Dqf)(x) = −1− qn

1− q(qx; q)n−1. (1.10)

–6–

Now recall that 0 < q < 1. If DqF = f and f is continuous then, for real a,

F (a)− F (0) = a (1− q)

∞∑

k=0

f(aqk) qk. (1.11)

This suggests the definition of the q-integral

∫ a

0

f(x) dqx := a (1− q)

∞∑

k=0

f(aqk) qk. (1.12)

Note that it can be viewed as an infinite Riemann sum with nonequidistant mesh widths.In the limit, as q ↑ 1, the right hand side of (1.12) will tend to the classical integral∫ a

0f(x) dx.From (1.11) we can also obtain F (a) − F (b) expressed in terms of f . This suggests

the definition ∫ b

a

f(x) dqx :=

∫ a

0

f(x) dqx−

∫ b

0

f(x) dqx. (1.13)

Note that (1.12) and (1.13) remain valid if a or b is negative.There is no unique canonical choice for the q-integral from 0 to ∞. We will put

∫ ∞

0

f(x) dqx := (1− q)

∞∑

k=−∞

f(qk) qk

(provided the sum converges absolutely). The other natural choices are then expressed by

a

∫ ∞

0

f(ax) dqx = a (1− q)∞∑

k=−∞

f(aqk) qk, a > 0.

Note that the above expression remains invariant when we replace a by aqn (n ∈ Z).As an example consider

∫ 1

0

xα dqx = (1− q)

∞∑

k=0

qk(α+1) =1− q

1− qα+1, Reα > −1, (1.14)

which tends, for q ↑ 1, to1

α+ 1=

∫ 1

0

xα dx. (1.15)

From the point of view of explicitly computing definite integrals by Riemann sum ap-proximation this is much more efficient than the approximation with equidistant meshwidths

1

n

n∑

k=1

(k

n

)α

.

The q-approximation (1.14) of the definite integral (1.15) (viewed as an area) goes essen-tially back to Fermat.

–7–

1.3. Elementary examples. The q-binomial series is defined by

1φ0(a;−; q, z) :=∞∑

k=0

(a; q)k(q; q)k

zk, |z| < 1. (1.16)

Here the ‘−’ in a rφs expression denotes an empty parameter list. The name “q-binomial”is justified since (1.16), with a replaced by qa, formally tends, as q ↑ 1, to the binomialseries

1F0(a; z) :=

∞∑

k=0

(a)kk!

zk = (1− z)−a, |z| < 1. (1.17)

The limit transition is made rigorous in [22, Appendix A]. It becomes elementary in thecase of a terminating series (a := q−n in (1.16) and a := −n in (1.17), where n ∈ Z+).

The q-analogue of the series evaluation in (1.17) is as follows.

Proposition 1.2 We have

1φ0(a;−; q, z) =(az; q)∞(z; q)∞

, |z| < 1. (1.18)

In particular,

1φ0(q−n;−; q, z) = (q−nz; q)n . (1.19)

Proof Put ha(z) := 1φ0(a;−; q, z). Then

(1− z) ha(z) = (1− az) ha(qz), hence ha(z) =(1− az)

(1− z)ha(qz).

Iteration gives

ha(z) =(az; q)n(z; q)n

ha(qnz), n ∈ Z+.

Now use that ha is analytic and therefore continuous at 0 and use that ha(0) = 1. Thus,for n→∞ we obtain (1.18).

The two most elementary q-hypergeometric series are the two q-exponential series

eq(z) := 1φ0(0;−; q, z) =

∞∑

k=0

zk

(q; q)k=

1

(z; q)∞, |z| < 1, (1.20)

and

Eq(z) := 0φ0(−;−; q,−z) =

∞∑

k=0

qk(k−1)/2 zk

(q; q)k= (−z; q)∞, z ∈ C. (1.21)

The evaluation in (1.20) is a specialization of (1.18). On the other hand, (1.21) is aconfluent limit of (1.18):

lima→∞

1φ0(a;−; q,−z/a) = 0φ0(−;−; q,−z)

–8–

(cf. (1.5)). The limit is uniform for z in compact subsets of C, see Lemma 1.1. Thus theevaluation in (1.21) follows also from (1.18).

It follows from (1.20) and (1.21) that

eq(z)Eq(−z) = 1. (1.22)

This identity is a q-analogue of ez e−z = 1. Indeed, the two q-exponential series areq-analogues of the exponential series by the limit formulas

limq↑1

Eq((1− q)z) = ez = limq↑1

eq((1− q)z).

The first limit is uniform on compacta of C by the majorization

∣∣∣∣qk(k−1)/2 (1− q)k zk

(q; q)k

∣∣∣∣ ≤|z|k

k!.

The second limit then follows by use of (1.22).Although we assumed the convention 0 < q < 1, it is often useful to find out for

a given q-hypergeometric series what will be obtained by changing q into q−1 and thenrewriting things again in base q. This will establish a kind of duality for q-hypergeometricseries. For instance, we have

eq−1(z) = Eq(−qz),

which can be seen from the power series definitions.The following four identities, including (1.20), are obtained from each other by trivial

rewriting.

∞∑

k=0

(1− q)k zk

(q; q)k=

1

((1− q)z; q)∞,

∞∑

k=0

zk

(q; q)k=

1

(z; q)∞,

(1− q)1−b∞∑

k=0

qkb(qk+1; q)∞ =(q; q)∞

(qb; q)∞ (1− q)b−1,

∫ (1−q)−1

0

tb−1 ((1− q)qt; q)∞ dqt =(q; q)∞

(qb; q)∞ (1− q)b−1, Re b > 0. (1.23)

As q ↑ 1, the first identity tends to

∞∑

k=0

zk

k!= ez,

while the left hand side of (1.23) tends formally to

∫ ∞

0

tb−1 e−t dt.

–9–

Since this last integral can be evaluated as Γ(b), it is tempting to consider the right handside of (1.23) as a the q-gamma function. Thus we put

Γq(z) :=(q; q)∞

(qz; q)∞ (1− q)z−1=

∫ (1−q)−1

0

tz−1Eq(−(1− q)qt) dqt, Re z > 0.

where the last identity follows from (1.23). It was proved in [22, Appendix B] that

limq↑1

Γq(z) = Γ(z), z 6= 0,−1,−2, . . . . (1.24)

We have just seen an example how an identity for q-hypergeometric series can havetwo completely different limit cases as q ↑ 1. Of course, this is achieved by differentrescaling. In particular, reconsideration of a power series as a q-integral is often helpfuland suggestive for obtaining distinct limits.

Regarding Γq it can yet be remarked that it satisfies the functional equation

Γq(z + 1) =1− qz

1− qΓq(z)

and that

Γq(n+ 1) =(q; q)n(1− q)n

, n ∈ Z+.

Similarly to the chain of equivalent identities including (1.20) we have a chain including(1.18):

∞∑

k=0

(qa; q)k zk

(q; q)k=

(qaz; q)∞(z; q)∞

,

(1− q)

∞∑

k=0

qkb(qk+1; q)∞(qk+a; q)∞

=(1− q) (q, qa+b; q)∞

(qa, qb; q)∞,

∫ 1

0

tb−1 (qt; q)∞(qat; q)∞

dqt =Γq(a) Γq(b)

Γq(a+ b), Re b > 0. (1.25)

In the limit, as q ↑ 1, the first identity tends to

∞∑

k=0

(a)k zk

k!= (1− z)−a,

while (1.25) formally tends to

∫ 1

0

tb−1 (1− t)a−1 dt =Γ(a) Γ(b)

Γ(a+ b). (1.26)

Thus (1.25) can be considered as a q-beta integral and we define the q-beta function by

Bq(a, b) :=Γq(a) Γq(b)

Γq(a+ b)=

(1− q) (q, qa+b; q)∞(qa, qb; q)∞

=

∫ 1

0

tb−1 (qt; q)∞(qat; q)∞

dqt. (1.27)

–10–

1.4. Heine’s 2φ1 series. Euler’s integral representation for the 2F1 hypergeometricfunction

2F1(a, b; c; z) =Γ(c)

Γ(b)Γ(c− b)

∫ 1

0

tb−1 (1− t)c−b−1 (1− tz)−a dt,

Re c > Re b > 0, | arg(1− z)| < π,

(1.28)

(cf. [13, 2.1(10)]) has the following q-analogue due to Heine.

2φ1(qq, qb; qc; q, z) =

Γq(c)

Γq(b)Γq(c− b)

∫ 1

0

tb−1 (tq; q)∞(tqc−b; q)∞

(tzqa; q)∞(tz; q)∞

dqt,

Re b > 0, |z| < 1.

(1.29)

Note that the left hand side and right hand side of (1.29) tend formally to the corre-sponding sides of (1.28). The proof of (1.29) is also analogous to the proof of (1.28).Expand (tzqa; q)∞/(tz; q)∞ as a power series in tz by (1.16), interchange summation andq-integration, and evaluate the resulting q-integrals by (1.25).

If we rewrite the q-integral in (1.29) as a series according to the definition (1.12), andif we replace qa, qb, qc by a, b, c then we obtain the following transformation formula:

2φ1(a, b; c; q, z) =(az; q)∞(z; q)∞

(b; q)∞(c; q)∞

2φ1(c/b, z; az; q, b). (1.30)

Although (1.29) and (1.30) are equivalent, they look quite different. In fact, in its form(1.30) the identity has no classical analogue. We see a new phenomenon, not occurring for

2F1, namely that the independent variable z of the left hand side mixes on the right handside with the parameters. So, rather than having a function of z with parameters a, b, c,we deal with a function of a, b, c, z satisfying certain symmetries.

Just as in the classical case (cf. [13, 2.1(14)]), substitution of some special value of z inthe q-integral representation (1.29) reduces it to a q-beta integral which can be explicitlyevaluated. We obtain

2φ1(a, b; c; q, c/(ab)) =(c/a, c/b; q)∞(c, c/(ab); q)∞

, |c/(ab)| < 1, (1.31)

where the more relaxed bounds on the parameters are obtained by analytic continuation.The terminating case of (1.31) is

2φ1(q−n, b; c; q, cqn/b) =

(c/b; q)n(c; q)n

, n ∈ Z+. (1.32)

The two fundamental transformation formulas

2F1(a, b; c; z) = (1− z)−a2F1(a, c− b; c; z/(z − 1)) (1.33)

= (1− z)c−a−b2F1(c− a, c− b; c; z)

–11–

(cf. [13, 2.1(22) and (23)]) have the following q-analogues.

2φ1(a, b; c; q, z) =(az; q)∞(z; q)∞

2φ2(a, c/b; c, az; q, bz) (1.34)

=(abz/c; q)∞(z; q)∞

2φ1(c/a, c/b; c; q, abz/c). (1.35)

Formula (1.35) can be proved either by threefold iteration of (1.30) or by twofold iterationof (1.34). The proof of (1.34) is more involved. Write both sides as power series in z. Thenmake both sides into double series by substituting for (b; q)k/(c; q)k on the left hand side aterminating 2φ1 (cf. (1.32)) and by substituting for (aqkz; q)∞/(z; q)∞ on the right handside a q-binomial series (cf. (1.16)). The result follows by some rearrangement of series.See [15, §1.5] for the details.

Observe the difference between (1.33) and its q-analogue (1.34). The argument z/(z−1) in (1.33) no longer occurs in (1.34) as a rational function of z, but the z-variable isdistributed over the argument and one of the lower parameters of the 2φ2. Also we do notstay within the realm of 2φ1 functions.

Equation (1.8) for s := 1 becomes

2φ1(q−n, b; c; q, z) = q−n(n+1)/2 (b; q)n

(c; q)n(−z)n 2φ1

(q−n,

q−n+1

c;q−n+1

b; q,

qn+1c

bz

),

n ∈ Z+.(1.36)

We may apply (1.36) to the preceding evaluation and transformation formulas for 2φ1 inorder to obtain new ones in the terminating case. From (1.32) we obtain

2φ1(q−n, b; c; q, q) =

(c/b; q)n bn

(c; q)n, n ∈ Z+. (1.37)

By inversion of direction of summation on both sides of (1.34) (with a := q−n) we obtain

2φ1(q−n, b; c; q, z) =

(c/b; q)n(c; q)n

3φ2

[q−n, b, bzq−n/c

bq1−n/c, 0; q, q

], n ∈ Z+. (1.38)

A terminating 2φ1 can also be transformed into a terminating 3φ2 with one of theupper parameters zero (result of Jackson):

2φ1(q−n, b; c; q, z) = (q−nbz/c; q)n 3φ2

[q−n, c/b, 0

c, cqb−1z−1; q, q

](1.39)

This formula can be proved by applying (1.19) to the factor (cqk+1b−1z−1; q)n−k occuringin the kth term of the right hand side. Then interchange summation in the resulting doublesum and substitute (1.37) for the inner sum. See [21, p.101] for the details.

–12–

1.5. A three-term transformation formula. Formula (1.36) has the following gen-eralization for non-terminating 2φ1:

2φ1(a, b; c; q, z) +(a, q/c, c/b, bz/q, q2/bz; q)∞

(c/q, aq/c, q/b, bz/c, cq/(bz); q)∞2φ1

(aq

c,bq

c;q2

c; q, z

)

=(abz/c, q/c, aq/b, cq/(abz); q)∞(bz/c, q/b, aq/c, cq/(bz); q)∞

2φ1

(a,aq

c;aq

b; q,

cq

abz

),

∣∣∣cq

ab

∣∣∣ < |z| < 1. (1.40)

This identity is a q-analogue of [13, 2.1(17)]. In the following proposition we rewrite (1.40)in an equivalent form and next sketch the elegant proof due to [28].

Proposition 1.3 Suppose {qn | n ∈ Z+} is disjoint from {a−1q−n, b−1q−n | n ∈ Z+}.Then

(z, qz−1; q)∞ 2φ1(a, b; c; q, z) = (az, qa−1z−1; q)∞(c/a, b; q)∞(c, b/a; q)∞

×2φ1(a, qa/c; qa/b; q, qc/(abz))+ (a←→ b). (1.41)

Proof Consider the function

F (w) :=(a, b, cw, q, qwz−1, w−1z; q)∞

(aw, bw, c, w−1; q)∞

1

w.

Its residue at qn (n ∈ Z+) equals the nth term of the series on the left hand side of (1.41).

The negative of its residue at a−1q−n (n ∈ Z+) equals the nth term of the first series onthe right hand side of (1.41), and the residue at b−1q−n is similarly related to the secondseries on the right hand side. Let C be a positively oriented closed curve around 0 in C

which separates the two sets mentioned in the Proposition. Then (2πi)−1∫CF (w) dw can

be expressed in two ways as an infinite sum of residues: either by letting the contour shrinkto {0} or by blowing it up to {∞}.

If we put z := q in (1.40) and substitute (1.31) then we obtain a generalization of(1.37) for non-terminating 2φ1:

2φ1(a, b; c; q, q) +(a, b, q/c; q)∞

(aq/c, bq/c, c/q; q)∞2φ1

(aq

c,bq

c;q2

c; q, q

)=

(abq/c, q/c; q)∞(aq/c, bq/c; q)∞

. (1.42)

By (1.13) this identity (1.42) can be equivalently written in q-integral form:

∫ 1

qc−1

(ct, qt; q)∞(at, bt; q)∞

dqt =(1− q) (abq/c, q/c, c, q; q)∞

(aq/c, bq/c, a, b; q)∞. (1.43)

Replace in (1.43) c, a, b by −qc, −qa, qb, respectively, and let q ↑ 1. Then we obtainformally ∫ 1

−1

(1 + t)a−c (1− t)b−1 dt =2b+a−c Γ(a− c+ 1) Γ(b)

Γ(a+ b− c+ 1). (1.44)

Note that, although it is trivial to obtain (1.44) from (1.26) by an affine transformationof the integration variable, their q-analogues (1.25) and (1.43) are by no means triviallyequivalent.

–13–

1.6. Bilateral series. Put a := q in (1.40) and substitute (1.16). Then we can combinethe two series from 0 to ∞ into a series from −∞ to ∞ without “discontinuity” in thesummand:

∞∑

k=−∞

(b; q)k(c; q)k

zk =(q, c/b, bz, q/(bz); q)∞(c, q/b, z, c/(bz); q)∞

, |c/b| < |z| < 1. (1.45)

Here we have extended the definition of (b; q)k by

(b; q)k :=(b; q)∞(bqk; q)∞

, k ∈ Z.

Formula (1.45) was first obtained by Ramanujan (Ramanujan’s 1ψ1-summation formula).It reduces to the q-binomial formula (1.18) for c := qn (n ∈ Z+). In fact, this observationcan be used for a proof of (1.45), cf. [17], [2, Appendix C]. Formula (1.45) is a q-analogueof the explicit Fourier series evaluation

∞∑

k=−∞

(b)k(c)k

eikθ =Γ(c) Γ(1− b)

Γ(c− b)ei(1−c)(θ−π) (1− eiθ)c−b−1,

0 < θ < 2π, Re (c− b− 1) > 0,

where (b)k := Γ(b+ k)/Γ(b), also for k = −1,−2, . . . .Define bilateral q-hypergeometric series by

rψs

[a1, . . . , arb1, . . . , bs

; q, z

]

:=∞∑

k=−∞

(a1, . . . , ar; q)k(b1, . . . , bs; q)k

((−1)k qk(k−1)/2

)s−rzk (1.46)

=r+1φs

[a1, . . . , ar, q

b1, . . . , bs; q, z

]

+(b1 − q) . . . (bs − q)

(a1 − q) . . . (ar − q) zs+1φs

[q2/b1, . . . , q

2/bs, q

q2/a1, . . . , q2/ar, 0, . . . , 0; q,

b1 . . . bsa1 . . . arz

], (1.47)

where a1, . . . ar, b1, . . . , bs 6= 0 and s ≥ r. The Laurent series in (1.46) is convergent for∣∣∣∣b1 . . . bsa1 . . . ar

∣∣∣∣ < |z| if s > r

and for ∣∣∣∣b1 . . . bsa1 . . . ar

∣∣∣∣ < |z| < 1 if s = r.

If some of the lower parameters in the rψs are 0 then a suitable confluent limit has to betaken in the s+1φs of (1.47).

Thus we can write (1.45) as

1ψ1(b; c; q, z) =(q, c/b, bz, q/(bz); q)∞(c, q/b, z, c/(bz); q)∞

, |c/b| < |z| < 1. (1.48)

–14–

Replace in (1.48) z by z/b, substitute (1.47), let b→∞, and apply (1.6) Then we obtain

0ψ1(−; c; q, z) :=

∞∑

k=−∞

(−1)k qk(k−1)/2 zk

(c; q)k=

(q, z, q/z; q)∞(c, c/z; q)∞

, |z| > |c|. (1.49)

In particular, for c = 0, we get the Jacobi triple product identity

0ψ1(−; 0; q, z) :=∞∑

k=−∞

(−1)k qk(k−1)/2 zk = (q, z, q/z; q)∞ , z 6= 0. (1.50)

The series in (1.50) is essentially a theta function. With the notation [14, 13.19(9)]we get

θ4(x; q) :=

∞∑

k=−∞

(−1)k qk2

e2πikx

=0ψ1(−; 0; q2, q e2πix)

=(q2, q e2πix, q e−2πix; q2)∞

=∞∏

k=1

(1− q2k) (1− 2q2k−1 cos(2πx) + q4k−2),

and similarly for the other theta functions θi(x; q) (i = 1, 2, 3).

1.7. The q-hypergeometric q-difference equation. Just as the hypergeometric dif-

ferential equation

z (1− z) u′′(z) + (c− (a+ b+ 1)z) u′(z)− a b u(z) = 0

(cf. [13, 2.1(1)]) has particular solutions

u1(z) := 2F1(a, b; c; z), u2(z) := z1−c2F1(a− c+ 1, b− c+ 1; 2− c; z),

the q-hypergeometric q-difference equation

z (qc − qa+b+1z) (D2qu)(z) +

[1− qc

1− q−

(qb

1− qa

1− q+ qa

1− qb+1

1− q

)z

](Dqu)(z)

−1− qa

1− q

1− qb

1− qu(z) = 0 (1.51)

has particular solutions

u1(z) := 2φ1(qa, qb; qc; q, z), (1.52)

u2(z) := z1−c2φ1(q

1+a−c, q1+b−c; q2−c; q, z). (1.53)

There is an underlying theory of q-difference equations with regular singularities, similarlyto the theory of differential equations with regular singularities discussed for instance inOlver [30, Ch. 5]. It is not difficult to prove the following proposition.

–15–

Proposition 1.4 Let A(z) :=∑∞

k=0 ak zk and B(z) :=

∑∞k=0 bk z

k be convergent powerseries. Let λ ∈ C be such that

(1− qλ+k) (1− qλ+k−1)

(1− q)2+ a0

1− qλ+k

1− q+ b0

{= 0, k = 0,6= 0, k = 1, 2, . . . .

(1.54)

Then the q-difference equation

z2 (D2qu)(z) + z A(z) (Dqu)(z) +B(z) u(z) = 0 (1.55)

has an (up to a constant factor) unique solution of the form

u(z) =∞∑

k=0

ck zλ+k. (1.56)

Note that (1.51) can be rewritten in the form (1.55) with a0 = (q−c − 1)/(1 − q),b0 = 0, so (1.54) has solutions λ = 0 and −c + 1 (mod (2πi log q−1)Z) provided c /∈ Z

(mod (2πi log q−1)Z). For the coefficients ck in (1.56) we find the recursion

ck+1

ck=

(1− qa+λ+k) (1− qb+λ+k)

(1− qc+λ+k) (1− qλ+k+1).

Thus we obtain solutions u1, u2 as given in (1.52), (1.53).Proposition 1.4 can also be applied to the case z = ∞ of (1.51). Just make the

transformation z 7→ z−1 in (1.51). One solution then obtained is

u3(z) := z−a2φ1

[qa, qa−c+1

qa−b+1; q, q−a−b+c+1 z−1

].

Now (1.40) can be rewritten as

u1(z) +(qa, q1−c, qc−b; q)∞

(qc−1, qa−c+1, q1−b; q)∞

(qb−1z, q2−bz−1; q)∞ zc−1

(qb−cz, qc−b+1z−1; q)∞u2(z)

=(q1−c, qa−b+1; q)∞(q1−b, qa−c+1; q)∞

(qa+b−cz, qc−a−b+1z−1; q)∞ za

(qb−cz, qc−b+1z−1; q)∞u3(z). (1.57)

Note that u3 is not a linear combination of u1 and u2, as the coefficients of u2 and u3 in(1.57) depend on z. However, since an expression of the form

z 7→(qαz, q1−αz−1; q)∞ zα−β

(qβz, q1−βz−1; q)∞

is invariant under transformations z 7→ qz, each term in (1.57) is a solution of (1.51). Thuseverything works fine when we restrict ourselves to a subset of the form {z0 q

k | k ∈ Z}.The q-analogue of the regular singularity at z = 1 for the ordinary hypergeometric

differential equation has to be treated in a different way. We can rewrite (1.51) as

(qc − qa+bz) u(qz) + (−qc − q + (qa + qb)z) u(z) + (q − z) u(q−1z) = 0.

It can be expected that the points z = qc−a−b and z = q, where the coefficient of u(qz)respectively u(q−1z) vanishes, will replace the classical regular singularity z = 1, see also(1.31), (1.37) and (1.42). A systematic theory for such singularities has not yet beendeveloped.

–16–

1.8. q-Bessel functions. The classical Bessel function is defined by

Jν(z) :=(z/2)ν∞∑

k=0

(−1)k

Γ(ν + k + 1) k!(z/2)2k

=(z/2)ν

Γ(ν + 1)0F1(−; ν + 1;−z2/4),

cf. [14, Ch. 7]). Jackson (1905) introduced two q-analogues of the Bessel function:

J (1)ν (z; q) :=

(qν+1; q)∞(q; q)∞

(z/2)ν 2φ1(0, 0; qν+1; q,−z2/4)

=(z/2)ν∞∑

k=0

(qν+k+1; q)∞(q; q)∞

(−1)k (z/2)2k

(q; q)k, (1.58)

J (2)ν (z; q) :=

(qν+1; q)∞(q; q)∞

(z/2)ν 0φ1(−; qν+1; q,−qν+1z2/4)

=(z/2)ν∞∑

k=0

(qν+k+1; q)∞(q; q)∞

qk(k+ν) (−1)k (z/2)2k

(q; q)k. (1.59)

Formally we havelimq↑1

J (i)ν ((1− q)z; q) = Jν(z), i = 1, 2

(cf. (1.24)). The two q-Bessel functions J(i)ν (z; q) can be simply expressed in terms of each

other. From (1.35) and (1.6) we obtain

2φ1(0, b; c; q, z) =1

(z; q)∞1φ1(c/b; c; q, bz).

A further confluence with b→ 0 yields

2φ1(0, 0; c; q, z) =1

(z; q)∞0φ1(−; c; q, cz).

This can be rewritten as

J (2)ν (z; q) = (−z2/4; q)∞ J (1)

ν (z; q).

Yet another q-analogue of the Bessel function is as follows.

Jν(z; q) :=(qν+1; q)∞(q; q)∞

zν 1φ1(0; qν+1; q, qz2)

=zν∞∑

k=0

(qν+k+1; q)∞(q; q)∞

(−1)k qk(k+1)/2 z2k

(q; q)k. (1.60)

Formally we havelimq↑1

Jν((1− q1/2)z; q) = Jν(z).

This so-called Jackson’s third q-Bessel function is not simply related to the q-Bessel func-tions (1.58), (1.59), but they were all introduced by Jackson. Koornwinder & Swarttouw[24] gave a satisfactory q-analogue of the Hankel transform in terms of the q-Bessel function(1.60).

–17–

1.9. Various results. Goursat’s list of quadratic transformations for Gaussian hyper-geometric functions can be found in [13, §2.11]. In a recent paper Rahman & Verma [31]have given a full list of q-analogues of Goursat’s table. However, all their formulas involveon at least one of both sides an 8φ7 series. Moreover, a good foundation from the theory ofq-difference equations is not yet available. For terminating series many of their 8φ7’s willsimplify to 4φ3’s. In section 2 we will meet some natural examples of these transformationscoming from orthogonal polynomials.

An important part of the book by Gasper & Rahman [15] deals with the derivation ofsummation and transformation formulas of s+1φs functions with s > 1. A simple exampleis the q-Saalschutz formula

3φ2(a, b, q−n; c, abc−1q1−n; q, q) =

(c/a, c/b; q)n(c, c/(ab); q)n

, n ∈ Z+, (1.61)

which follows easily from (1.35) by expanding the quotient on the right hand side of (1.35)with the aid of (1.18), and next comparing equal powers of z at both sides of (1.35).Formula (1.61) is the q-analogue of the Pfaff-Saalschutz formula

3F2(a, b,−n; c, 1+ a+ b− c− n; 1) =(c− a)n (c− b)n(c)n (c− a− b)n

, n ∈ Z+,

which can be proved in an analogous way as (1.61).An example of a much more involved transformation formula, which has many impor-

tant special cases, is Watson’s transformation formula

8φ7

[a, qa1/2,−qa1/2, b, c, d, e, q−n

a1/2,−a1/2, aq/b, aq/c, aq/d, aq/e, aqn+1; q,

a2q2+n

bcde

]

=(aq, aq/(de); q)n(aq/d, aq/e; q)n

4φ3

[q−n, d, e, aq/(bc)

aq/b, aq/c, deq−n/a; q, q

], n ∈ Z+ , (1.62)

cf. Gasper & Rahman [15, §2.5]. Then, for a2qn+1 = bcde, the right hand side can beevaluated by use of (1.61). The resulting evaluation

(aq, aq/(bc), aq/(bd), aq/(cd); q)n(aq/b, aq/c, aq/d, aq/(bcd); q)n

of the left hand side of (1.62) subject to the given relation between a, b, c, d, e, q is calledJackson’s summation formula, cf. [15, §2.6].

The famous Rogers-Ramanujan identities

0φ1(−; 0; q, q) :=∞∑

k=0

qk2

(q; q)k=

(q2, q3, q5; q5)∞(q; q)∞

,

0φ1(−; 0; q, q2) :=

∞∑

k=0

qk(k+1)

(q; q)k=

(q, q4, q5; q5)∞(q; q)∞

have been proved in many different ways (cf. Andrews [2]), not only analytically but alsoby an interpretation in combinatorics or in the framework of Kac-Moody algebras. A quickanalytic proof starting from (1.62) is described in [15, §2.7].

–18–

Exercises to §1

1.1 Prove that(a; q)n = (−a)n qn(n−1)/2 (a−1q1−n; q)n.

1.2 Prove that

(a; q)2n = (a; q2)n (aq; q2)n,

(a2; q2)n = (a; q)n (−a; q)n,

(a; q)∞ = (a1/2,−a1/2, (aq)1/2,−(aq)1/2; q)∞.

1.3 Prove the following identity of Euler:

(−q; q)∞ (q; q2)∞ = 1.

1.4 Prove that∞∑

l=−∞

(−1)l−m q(l−m)(l−m−1)/2

Γq(n− l + 1) Γq(l −m+ 1)= δn,m, 0 < q ≤ 1.

Do it first for q = 1. Start for instance with ez e−z = 1 and, in the q-case, witheq(z)Eq(−z) = 1.

1.5 Let1

(q; q)∞=

∞∑

n=0

an qn

be the power series expansion of the left hand side in terms of q. Show that an equals thenumber of partitions of n.Show also that the coefficient bk,n in

qk

(q; q)k=

∞∑

n=k

bk,n qn

is the number of partitions of n with highest part k. Give now a partition theoretic proofof the identity

1

(q; q)∞=

∞∑

k=0

qk

(q; q)k.

1.6 In the same way, give a partition theoretic proof of the identity

1

(qm; q)∞=

∞∑

k=0

qmk

(q; q)k, m ∈ N.

1.7 Give also a partition theoretic proof of

∞∑

k=0

qk(k−1)/2qmk

(q; q)k= (−qm; q)∞, m ∈ N.

(Consider the problem first for m = 1.)

–19–

1.8 Let GF (p) be the finite field with p elements. Let A be a n× n matrix with entrieschosen independently and at random from GF (p), with equal probability for the fieldelements to be chosen. Let q := p−1. Prove that the probability that A is invertible is(q; q)n. (See SIAM News 23 (1990) no.6, p.8.)

1.9 Let GF (p) be as in the previous exercise. Let V be an n-dimensional vector spaceover GF (p). Prove that the number of k-dimensional linear subspaces of V equals theq-binomial coefficient [n

k

]

p:=

(p; p)n(p; p)k (p; p)n−k

.

1.10 Show that

(ab; q)n =

n∑

k=0

[nk

]

qbk (a; q)k (b; q)n−k.

Show that both Newton’s binomial formula and the formula

(a+ b)n =n∑

k=0

(n

k

)(a)k (b)n−k

are limit cases of the above formula.

2. q-Analogues of the classical orthogonal polynomials

Originally by classical orthogonal polynomials were only meant the three families of Ja-cobi, Laguerre and Hermite polynomials, but recent insights consider a much bigger classof polynomials as “classical”. On the one hand, there is an extension to hypergeometricorthogonal polynomials up to the 4F3 level and including certain discrete orthogonal poly-nomials. These are brought together in the Askey tableau, cf. Table 1. On the other handthere are q-analogues of all the families in the Askey tableau, often several q-analoguesfor one classical family (cf. Table 2 for some of them). The master class of all these q-analogues is formed by the celebrated Askey-Wilson polynomials. They contain all otherfamilies described in this chapter as special cases or limit cases. Good references for thischapter are Andrews & Askey [4] and Askey & Wilson [8]. See Koekoek & Swarttouw [19]for a quite comprehensive list of formulas. See also Atakishiyev, Rahman & Suslov [10] fora somewhat different approach.

Some parts of this section contain surveys without many proofs. However, the subsec-tions 2.3 and 2.4 on big and little q-Jacobi polynomials and 2.5 and 2.6 on the Askey-Wilsonintegral and related polynomials are rather self-contained.

2.1. Very classical orthogonal polynomials. An introduction to the traditionalclassical orthogonal polynomials can be found, for instance, in [14, Ch. 10]. One possiblecharacterization is as systems of orthogonal polynomials {pn}n=0,1,2,... which are eigenfunc-tions of a second order differential operator not involving n with eigenvalues λn dependingon n:

a(x) p′′n(x) + b(x) p′n(x) + c(x) pn(x) = λn pn(x). (2.1)

–20–

Because we will extend the definition of classical orthogonal polynomials in this section, wewill call the orthogonal polynomials satisfying (2.1) very classical orthogonal polynomials.

The classification shows that, up to an affine transformation of the independent variable,the only cases are as follows.

Jacobi polynomials

P (α,β)n (x) :=

(α+ 1)nn!

2F1(−n, n+ α+ β + 1;α+ 1; (1− x)/2), α, β > −1, (2.2)

orthogonal on [−1, 1] with respect to the measure (1− x)α (1 + x)β dx;

Laguerre polynomials

Lαn(x) :=

(α+ 1)nn!

1F1(−n;α+ 1; x), α > −1, (2.3)

orthogonal on [0,∞) with respect to the measure xα e−x dx;

Hermite polynomials

Hn(x) := (2x)n 2F0(−n/2, (1− n)/2;−;−x−2), (2.4)

orthogonal on (−∞,∞) with respect to the measure e−x2

dx.In fact, Jacobi polynomials are the generic case here, while the other two classes are

limit cases of Jacobi polynomials:

Lαn(x) = lim

β→∞P (α,β)n (1− 2x/β), (2.5)

Hn(x) = 2n n! limα→∞

α−n/2 P (α,α)n (α−1/2x). (2.6)

Hermite polynomials are also limit cases of Laguerre polynomials:

Hn(x) = (−1)n 2n/2 n! limα→∞

α−n/2 Lαn((2α)

1/2x+ α). (2.7)

The limit (2.5) is immediate from (2.2) and (2.3). The limit (2.6) follows from (2.4) and asimilar series representation for Gegenbauer or ultraspherical polynomials (special Jacobipolynomials with α = β):

P (α,α)n (x) =

(α+ 1)n (α+ 1/2)n(2α+ 1)n n!

(2x)n 2F1(−n/2, (1− n)/2;−α− n+ 1/2; x−2),

cf. [14, 10.9(4) and (18)]. The limit (2.7) cannot be easily derived by comparison of seriesrepresentations. One method of proof is to rewrite the three term recurrence relation forLaguerre polynomials

(n+ 1)Lαn+1(x)− (2n+ α+ 1− x)Lα

n(x) + (n+ α)Lαn−1(x) = 0

–21–

(cf. [14, 10.12(8)]) in terms of the polynomials in x given by the right hand side of (2.7)and, next, to compare it with the three term recurrence relation for Hermite polynomials(cf. [14, 10.13(10)])

Hn+1(x)− 2xHn(x) + 2nHn−1(x) = 0.

Very classical orthogonal polynomials have other characterizations, for instance bythe existence of a Rodrigues type formula or by the fact that the first derivatives againform a system of orthogonal polynomials (cf. [14, §10.6]). Here we want to point out that,associated with the last two characterizations, there is a pair of differential recurrencerelations from which many of the basic properties of the polynomials can be easily derived.For instance, for Jacobi polynomials we have the pair

d

dxP (α,β)n (x) =

n+ α+ β + 1

2P

(α+1,β+1)n−1 (x), (2.8)

(1− x)−α (1 + x)−β d

dx

((1− x)α+1 (1 + x)β+1 P

(α+1,β+1)n−1 (x)

)= −2nP (α,β)

n (x). (2.9)

The differential operators in (2.8), (2.9) are called shift operators because of their param-eter shifting property. The differential operator in (2.8) followed by the one in (2.9) yieldsthe second order differential operator of which the Jacobi polynomials are eigenfunctions.If we would have defined Jacobi polynomials only by their orthogonality property, not bytheir explicit expression, then we would have already been able to derive (2.8), (2.9) up

to constant factors just by the remarks that the operators D− in (2.8) and D(α,β)+ in (2.9)

satisfy

∫ 1

−1

(D−f)(x) g(x) (1− x)α+1 (1 + x)β+1 dx

= −

∫ 1

−1

f(x) (D(α,β)+ g)(x) (1− x)α (1 + x)β dx (2.10)

and that D− sends polynomials of degree n to polynomials of degree n − 1, while D(α,β)+

sends polynomials of degree n− 1 to polynomials of degree n.The same idea can be applied again and again for the more general classical orthogonal

polynomials we will discuss in this section.It follows from (2.8), (2.9) and (2.10) that

∫ 1

−1

(P (α,β)n (x)

)2(1− x)α (1 + x)β dx = const.

∫ 1

−1

(1− x)α+n (1 + x)β+n dx,

where the constant on the right hand side can be easily computed. What is left for computa-tion is a beta integral, which of course is elementary. However, we emphasize this reductionof computation of quadratic norms of classical orthogonal polynomials to computation ofthe integral of the weight function with shifted parameter, because this phenomenon willalso return in the generalizations. Usually, the computation of the integral of the weightfunction is the only nontrivial part. On the other hand, if we have some deep evaluationof a definite integral with positive integrand, then it is worth to explore the polynomialsbeing orthogonal with respect to the weight function given by the integrand.

–22–

2.2. The Askey tableau. Similar to the classification discussed in §2.1, one can classifyall systems of orthogonal polynomials {pn} which are eigenfunctions of a second orderdifference operator:

a(x) pn(x+ 1) + b(x) pn(x) + c(x) pn(x− 1) = λn pn(x). (2.11)

Here the definition of orthogonal polynomials is relaxed somewhat. We include the pos-sibility that the degree n of the polynomials pn only takes the values n = 0, 1, . . . , N andthat the orthogonality is with respect to a positive measure having support on a set ofN + 1 points. Hahn [16] studied the q-analogue of this classification (cf. §2.5) and hepointed out how the polynomials satisfying (2.11) come out as limit cases for q ↑ 1 of hisclassification.

The generic case for this classification is given by the Hahn polynomials

Qn(x;α, β,N) :=3F2

[−n, n+ α+ β + 1,−x

α+ 1,−N; 1

]

=n∑

k=0

(−n)k (n+ α+ β + 1)k (−x)k(α+ 1)k (−N)k k!

. (2.12)

Here we assume n = 0, 1, . . . , N and we assume the notational convention that

rFs

[−n, a2, . . . , arb1, . . . , bs

; z

]:=

n∑

k=0

(−n)k (a2)k . . . (ar)k(b1)k . . . (bs)k k!

zk, n ∈ Z+, (2.13)

remains well-defined when some of the bi are possibly non-positive integer but ≤ −n. Forthe notation of Hahn polynomials and other families to be discussed in this subsection wekeep to the notation of Askey & Wilson [8, Appendix] and Labelle’s poster [25]. Thereone can also find further references.

Hahn polynomials satisfy orthogonality relations

N∑

x=0

Qn(x)Qm(x) ρ(x) = δn,m1

πn, n,m = 0, 1 . . . , N, (2.14)

where

ρ(x) :=

(N

x

)(α+ 1)x (β + 1)N−x

(α+ β + 2)N(2.15)

and

πn :=

(N

n

)2n+ α + β + 1

α+ β + 1

(α+ 1)n (α+ β + 1)n(β + 1)n (N + α+ β + 2)n

. (2.16)

We get positive weights ρ(x) (or weights of fixed sign) if α, β ∈ (−1,∞) ∪ (−∞,−N).The other orthogonal polynomials coming out of this classification are limit cases of

Hahn polynomials. We get the following families.

–23–

Krawtchouk polynomials

Kn(x; p,N) := 2F1(−n,−x;−N ; p−1), n = 0, 1, . . . , N, 0 < p < 1, (2.17)

with orthogonality measure having weights x 7→(Nx

)px (1− p)N−x on {0, 1, . . . , N};

Meixner polynomials

Mn(x; β, c) := 2F1(−n,−x; β; 1− c−1), 0 < c < 1, β > 0, (2.18)

with orthogonality measure having weights x 7→ (β)x cx/x! on Z+;

Charlier polynomials

Cn(x; a) := 2F0(−n,−x;−;−a−1), a > 0,

with orthogonality measure having weights x 7→ ax/x! on Z+.Note that Krawtchouk polynomials are Meixner polynomials (2.18) with β := q−N .

Krawtchouk and Meixner polynomials are limits of Hahn polynomials, while Charlier poly-nomials are limits of both Krawtchouk and Meixner polynomials. In a certain sense, thevery classical orthogonal polynomials are also contained in the class discussed here, sinceJacobi, Laguerre and Hermite polynomials are limits of Hahn, Meixner and Charlier poly-nomials, respectively. For instance,

P (α,β)n (1− 2x) =

(α+ 1)nn!

limN→∞

Qn(Nx;α, β,N).

See Table 1 (or rather a part of it) for a pictorial representation of these families and theirlimit transitions.

The Krawtchouk, Meixner and Charlier polynomials are self-dual, i.e., they satisfy

pn(x) = px(n), x, n ∈ Z+ or x, n ∈ {0, 1, . . . , N}.

Thus the orthogonality relations and dual orthogonality relations of these polynomialsessentially coincide and the second order difference equation (2.11) becomes the threeterm recurrence relation after interchange of x and n. Then it can be arranged that theeigenvalue λn in (2.11) becomes n. By the self-duality, the L2 completeness of the systemsin case N =∞ is also clear.

For Hahn polynomials we have no self-duality, so the dual orthogonal system will bedifferent. Observe that, in (2.12), we can write

(−n)k (n+ α+ β + 1)k =k−1∏

j=0

(−n(n+ α + β + 1) + j(j + α+ β + 1)

),

which is a polynomial of degree k in n(n+ α+ β + 1). Now define

Rn(x(x+ α + β + 1);α, β,N) := Qx(n;α, β,N), n, x = 0, 1, . . . , N.

–24–

Then Rn (n = 0, 1, . . . , N) extends to a polynomial of degree n:

Rn(x(x+ α+ β + 1);α, β,N) = 3F2

[−n,−x, x+ α+ β + 1

α + 1,−N; 1

]

=n∑

k=0

(−n)k (−x)k (x+ α+ β + 1)k(α+ 1)k (−N)k k!

.

These are called the dual Hahn polynomials. The dual orthogonality relations implied by(2.14) are the orthogonality relations for the dual Hahn polynomials:

N∑

x=0

Rm(x(x+ α+ β + 1))Rn(x(x+ α+ β + 1)) πx = δm,n1

ρ(n).

Here πx and ρ(n) are as in (2.15) and (2.16). Thus the dual Hahn polynomials are alsoorthogonal polynomials. The three term recurrence relation for Hahn polynomials trans-lates as a second order difference equation which is a slight generalization of (2.11). It hasthe form

a(x) pn(λ(x+ 1)) + b(x) pn(λ(x)) + c(x) pn(λ(x− 1)) = λn pn(λ(x)), (2.19)

where λ(x) := x(x+α+β+1) is a quadratic function of x. The Krawtchouk and Meixnerpolynomials are also limit cases of the dual Hahn polynomials.

It is a natural question to ask for other orthogonal polynomials being eigenfunctionsof a second order difference equation of the form (2.19). A four-parameter family with thisproperty are the Racah polynomials, essentially known for a long time to the physicists asRacah coefficients, which occur in connection with threefold tensor products of irreduciblerepresentations of the group SU(2). However, it was not recognized before the late seventies(cf. Wilson [33]) that orthogonal polynomials are hidden in the Racah coefficients. Racahpolynomials are defined by

Rn(x(x+ γ + δ + 1);α, β, γ, δ) := 4F3

[−n, n+ α+ β + 1,−x, x+ γ + δ + 1

α + 1, β + δ + 1, γ + 1; 1

], (2.20)

where α + 1 or β + δ + 1 or γ + 1 = −N for some N ∈ Z+, and where n = 0, 1, . . . , N .Similarly as for the dual Hahn polynomials it can be seen that the Racah polynomial Rn

is indeed a polynomial of degree n in λ(x) := x(x + γ + δ + 1). The Racah polynomialsare orthogonal with respect to weights w(x) on the points λ(x), x = 0, 1 . . . , N , given by

w(x) :=(γ + δ + 1)x ((γ + δ + 3)/2)x (α+ 1)x (β + δ + 1)x (γ + 1)xx! ((γ + δ + 1)/2)x (γ + δ − α+ 1)x (γ − β + 1)x (δ + 1)x

.

It is evident from (2.20) that dual Racah polynomials are again Racah polynomials withα, β interchanged with γ, δ. Hahn and dual Hahn polynomials can be obtained as limitcases of Racah polynomials.

Each orthogonality relation for the Racah polynomials is an explicit evaluation of afinite sum. It is possible to interpret this finite sum as a sum of residues coming from a

–25–

contour integral in the complex plane. This contour integral can also be considered andevaluated for values of N not necessarily in Z+. For suitable values of the parameters thecontour integral can then be deformed to an integral over the imaginary axis and it givesrise to the orthogonality relations for the Wilson polynomials (cf. Wilson [33]) defined by

Wn(x2; a, b, c, d)

:= (a+ b)n (a+ c)n (a+ d)n 4F3

[−n, n+ a+ b+ c+ d− 1, a+ ix, a− ix

a+ b, a+ c, a+ d; 1

]. (2.21)

Apparently, the right hand side defines a polynomial of degree n in x2. If a, b, c, d havepositive real parts and complex parameters appear in conjugate pairs then the functionsx 7→ Wn(x

2) (n ∈ Z+) of (2.21) are orthogonal with respect to the measure w(x)dx on[0,∞), where

w(x) :=

∣∣∣∣Γ(a+ ix) Γ(b+ ix) Γ(c+ ix) Γ(d+ ix)

Γ(2ix)

∣∣∣∣2

.

The normalization in (2.21) is such that the Wilson polynomials are symmetric in theirfour parameters a, b, c, d.

The Wilson polynomials satisfy an eigenfunction equation of the form

a(x)Wn((x+ i)2) + b(x)Wn(x2) + c(x)Wn((x− i)

2) = λnWn(x2).

So we have the new phenomenon that the difference operator at the left hand side shiftsinto the complex plane, out of the real interval on which the Wilson polynomials areorthogonal. This difference operator can be factorized as a product of two shift operatorsof similar type. They have properties and applications analogous to the shift operatorsfor Jacobi polynomials discussed at the end of §2.1. They also reduce the evaluation ofthe quadratic norms to the evaluation of the integral of the weight function, but this lastproblem is now much less trivial than in the Jacobi case.

Now, we can descend from the Wilson polynomials by limit transitions, just as we didfrom the Racah polynomials. On the 3F2 level we thus get continuous analogues of theHahn and dual Hahn polynomials as follows.

Continuous dual Hahn polynomials:

Sn(x2; a, b, c) := (a+ b)n (a+ c)n 3F2

[−n, a+ ix, a− ix

a+ b, a+ c; 1

],

where a, b, c have positive real parts; if one of these parameters is not real then one of theother parameters is its complex conjugate. The functions x 7→ Sn(x

2) are orthogonal withrespect to the measure w(x)dx on [0,∞), where

w(x) :=

∣∣∣∣Γ(a+ ix) Γ(b+ ix) Γ(c+ ix)

Γ(2ix)

∣∣∣∣2

.

Continuous Hahn polynomials:

pn(x; a, b, a, b) := in(a+ a)n (a+ b)n

n!3F2

[−n, n+ a+ a+ b+ b− 1, a+ ix

a+ a, a+ b; 1

],

–26–

where a, b have positive real part. The polynomials pn are orthogonal on R with respect tothe measure |Γ(a+ ix) Γ(b+ ix)|2 dx. In Askey & Wilson [8, Appendix] only the symmetriccase (a, b > 0 or a = b) of these polynomials occurs. The general case was discovered byAtakishiyev & Suslov [9].

Jacobi polynomials are limit cases of continuous Hahn polynomials and also directly ofWilson polynomials (with one pair of complex conjugate parameters). There is one furtherclass of orthogonal polynomials on the 2F1 level occurring as limit cases of orthogonalpolynomials on the 3F2 level:

Meixner-Pollaczek polynomials:

P (a)n (x;φ) :=

(2a)nn!

einφ 2F1(−n, a+ ix; 2a; 1− e−2iφ), a > 0, 0 < φ < π.

(Here we have chosen the normalization as in Labelle [25], which is the same as in Pol-laczek’s original 1950 paper.) They are orthogonal on R with respect to the weight functionx 7→ e(2φ−π)x |Γ(a+ ix)|2. They can be considered as continuous analogues of the Meixnerand Krawtchouk polynomials. They are limits of both continuous Hahn and continuousdual Hahn polynomials. Laguerre polynomials are limit cases of Meixner-Pollaczek poly-nomials. Note that the last three families are analytic continuations, both in x and in

–27–

the parameters, of dual Hahn polynomials, Hahn polynomials and Meixner polynomials,respectively.

All families of orthogonal polynomials discussed until now, together with the limittransitions between them, form the Askey tableau (or scheme or chart) of hypergeometric

orthogonal polynomials. See Askey & Wilson [8, Appendix], Labelle [25] or Table 1. Seealso [20] for group theoretic interpretations.

2.3. Big q-Jacobi polynomials. These polynomials were hinted at by Hahn [16] andexplicitly introduced by Andrews & Askey [4]. Here we will show how their basic propertiescan be derived from a suitable pair of shift operators. We keep the convention of §1 that0 < q < 1.

First we introduce q-integration by parts. This will involve backward and forward

q-derivatives:

(D−q f)(x) :=

f(x)− f(qx)

(1− q)x, (D+

q f)(x) :=f(q−1x)− f(x)

(1− q)x.

Here D−q coincides with Dq introduced in (1.9).

Proposition 2.1 If f and g are continuous on [−d, c] (c, d ≥ 0) then

∫ c

−d

(D−q f)(x) g(x) dqx = f(c) g(q−1c)− f(−d) g(−q−1d)−

∫ c

−d

f(x) (D+q g)(x) dqx.

Proof

∫ c

0

(D−q f)(x) g(x) dq(x) =

∞∑

k=0

(f(cqk)− f(cqk+1)

)g(cqk)

= limN→∞

{f(c) g(q−1c)− f(cqN+1) g(cqN) +

N∑

k=0

f(cqk)(g(cqk)− g(cqk−1)

)}

= f(c) g(q−1c)− f(0) g(0) +

∞∑

k=0

f(cqk)(g(cqk)− g(cqk−1)

)

= f(c) g(q−1c)− f(0) g(0)−

∫ c

0

f(x) (D+q g)(x) dqx.

Now apply (1.13).

Let

w(x; a, b, c, d; q) :=(qx/c,−qx/d; q)∞(qax/c,−qbx/d; q)∞

. (2.22)

Note that w(x; a, b, c, d; q) > 0 on [−d, c] if c, d > 0 and

[−c

dq< a <

1

q& −

d

cq< b <

1

q

]or

[a = cα & b = −dα for some α ∈ C\R.

](2.23)

–28–

From now on we assume that these inequalities hold. If f is continuous on [−d, c] then

limq↑1

∫ c

−d

f(x)w(x; qα, qβ, c, d; q) dqx =

∫ c

−d

f(x) (1− c−1x)α (1 + d−1x)β dx, (2.24)

Thus the measure w(x; a, b, c, d; q) dqx can be considered as a q-analogue of the Jacobipolynomial orthogonality measure shifted to an arbitrary finite interval containing 0.

Sincew(q−1c; a, b, c, d; q) = 0 = w(−q−1d; a, b, c, d; q),

we get from Proposition 2.1 that for any two polynomials f, g the following holds.

∫ c

−d

(D−q f)(x) g(x)w(x; qa, qb, c, d; q) dqx

= −

∫ c

−d

f(x)[D+

q

(g(.)w(.; qa, qb, c, d; q)

)](x) dqx. (2.25)

Define

(D+,a,bq f)(x) :=

[D+

q

(w(.; qa, qb, c, d; q) f(.)

)](x)

w(x; a, b, c, d; q)

=(1− q)−1 x−1(1− x/c) (1 + x/d) f(q−1x)

− (1− q)−1 x−1 (1− qax/c) (1 + qbx/d) f(x). (2.26)

In the followingD−q f(x) orD

−q (f(x)) will mean (D−

q f)(x). AlsoD+,a,bq f(x) orD+,a,b

q (f(x))

will mean (D+,a,bq f)(x). A simple computation yields the following.

D−q x

n =1− qn

1− qxn−1, (2.27)

D+,a,bq

((q2ax/c; q)n−1

)

=q−na−1 − qb

(1− q)d(qax/c; q)n + terms of degree ≤ n− 1 in x. (2.28)

Define the big q-Jacobi polynomial Pn(x; a, b, c, d; q) as the monic orthogonal polyno-mial of degree n in x with respect to the measure w(x; a, b, c, d; q) dqx on [−d, c]. Later wewill introduce another normalization for these polynomials and we will then write them asPn instead of Pn. It follows from (2.24) that

limq↑1

Pn(x; qα, qβ, c, d; q) = const. P (α,β)

n

(d− c+ 2x

d+ c

),

a Jacobi polynomial shifted to the interval [−d, c]. Big q-Jacobi polynomials with d = 0,c = 1 are called little q-Jacobi polynomials.

Two simple consequences of the definition are:

Pn(−x; a, b, c, d; q) = (−1)n Pn(x; b, a, d, c; q) (2.29)

and

–29–

Pn(λx; a, b, c, d; q) = λn Pn(x; a, b, λ−1c, λ−1d; q), λ > 0.

It follows from (2.25), (2.27) and (2.28) that D−q and D+,a,b

q act as shift operators on thebig q-Jacobi polynomials:

D−q Pn(x; a, b, c, d; q) =

1− qn

1− qPn−1(x; qa, qb, c, d; q), (2.30)

D+,a,bq Pn−1(x; qa, qb, c, d; q) =

q2ab− q−n+1

(1− q)cdPn(x; a, b, c, d; q). (2.31)

Composition of (2.30) and (2.31) yields a second order q-difference equation for the bigq-Jacobi polynomials:

D+,a,bq D−

q Pn(x; a, b, c, d; q) =q(1− q−n)(1− qn+1ab)

(1− q)2cdPn(x; a, b, c, d; q). (2.32)

The left hand side of (2.32) can be rewritten as

A(x)((D−

q )2Pn

)(q−1x) +B(x) (D−

q Pn)(q−1x)

= a(x)Pn(q−1x) + b(x)Pn(x) + c(x)Pn(qx) (2.33)

for certain polynomials A,B and a, b, c. Here

A(x) = q−1 (1− qax/c) (1 + qbx/d),

B(x) =(1− qb)c− (1− qa)d− (1− q2ab)x

(1− q)cd. (2.34)

Compare (2.32), (2.33) with (2.1), (2.11). Apparently we have here another extension ofthe concept of classical orthogonal polynomials. Two other properties point into the samedirection. First, by (2.30) the polynomials D−

q Pn+1 (n = 0, 1, 2, . . .) form again a systemof orthogonal polynomials. Second, iteration of (2.31) yields a Rodrigues type formula.

It follows from (2.26) that

(D+,a,bq f)

(c

qa

)= −

(1− qa) (1 + qad/c)

qad(1− q)f

(c

q2a

).

Combination with (2.31) yields the recurrence

q2ab− q−n+1

(1− q)cdPn

(c

qa; a, b, c, d; q

)

= −(1− qa)(1 + qad/c)

qad(1− q)Pn−1

(c

q2a; qa, qb, c, d; q

).

By iteration we get an evaluation of the big q-Jacobi polynomial at a special point:

Pn

(c

qa; a, b, c, d; q

)=

(c

qa

)n(qa; q)n (−qad/c; q)n

(qn+1ab; q)n. (2.35)

–30–

Now we will normalize the big q-Jacobi polynomials such that they take the value 1 atc/(qa):

Pn(x; a, b, c, d; q) :=Pn(x; a, b, c, d; q)

Pn(c/(qa); a, b, c, d; q). (2.36)

Note that the value at −d/(qb) now follows from (2.29) and (2.35):

Pn

(−d

qb; a, b, c, d; q

)=

(−ad

bc

)n(qb; q)n (−qbc/d; q)n(qa; q)n (−qad/c; q)n

. (2.37)

It follows from (2.29) and (2.37) that

Pn(−x; a, b, c, d; q) =

(−ad

bc

)n(qb; q)n (−qbc/d; q)n(qa; q)n (−qad/c; q)n

Pn(x; b, a, d, c; q).

The following lemma, proved by use of (1.10), gives the q-analogue of a Taylor seriesexpansion.

Lemma 2.2 If f(x) :=∑n

k=0 ck (qax/c; q)k then

((D−

q )kf

)( c

qk+1a

)= ck (−1)

k(qac

)k

qk(k−1)/2 (q; q)k(1− q)k

.

Now put f(x) := Pn(x; a, b, c, d; q) in Lemma 2.2 and substitute (2.30) (iterated)and (2.36). Then the ck can be found explicitly and we obtain a representation by aq-hypergeometric series:

Pn(x; a, b, c, d; q) =n∑

k=0

(q−n; q)k (qn+1ab; q)k (qax/c; q)k q

k

(qa; q)k (−qad/c; q)k (q; q)k

= 3φ2

[q−n, qn+1ab, qax/c

qa,−qad/c; q, q

]. (2.38)

Andrews & Askey [4, (3,28)] use the notation P(α,β)n (x; c, d : q), which coincides, up to a

constant factor, with our pn(x; qα, qβ, c, d; q).

It follows by combination of (2.29) and (2.38) that

Pn(x; a, b, c, d; q)

Pn(−d/(qb); a, b, c, d; q)= 3φ2

[q−n, qn+1ab,−qbx/d

qb,−qbc/d; q, q

], (2.39)

where the denominator of the left hand side is explicitly given by (2.37).Next we will compute the quadratic norms. By (2.25), (2.30) and (2.31) we get the

recurrence∫ c

−d

(P 2nw)(x; a, b, c, d; q) dqx =

qn−1 (1− qn) cd

1− qn+1ab

∫ c

−d

(P 2n−1w)(x; qa, qb, c, d; q) dqx

=qn(n−1)/2 (q; q)n (cd)

n

(qn+1ab; q)n

∫ c

−d

w(x; qna, qnb, c, d; q) dqx, (2.40)

–31–

where the second equality follows by iteration. Now the q-integral of w from −d to c canbe rewritten as a sum of two 2φ1’s of argument q of the form of the left hand side of (1.42).Evaluation by (1.42) yields

∫ c

−d

w(x; a, b, c, d; q) dqx = (1− q)c(q,−d/c,−qc/d, q2ab; q)∞(qa, qb,−qbc/d,−qad/c; q)∞

. (2.41)

Together with (2.40) this yields:

∫ c

−d(P 2

nw)(x; a, b, c, d; q) dqx∫ c

−dw(x; a, b, c, d; q) dqx

= qn(n−1)/2 (cd)n(q, qa, qb,−qbc/d,−qad/c; q)n

(q2ab; q)2n (qn+1ab; q)n. (2.42)

Now we can compute the coefficients in the three term recurrence relation for the bigq-Jacobi polynomials Pn(x) := Pn(x; a, b, c, d; q):

x Pn(x) = Pn+1(x) +Bn Pn(x) + Cn Pn−1(x). (2.43)

Then Cn is the quotient of the right hand sides of (2.42) for degree n and for degree n− 1,respectively, so

Cn =qn−1 (1− qn) (1− qna) (1− qnb) (1− qnab) (d+ qnbc) (c+ qnad)

(1− q2n−1ab) (1− q2nab)2 (1− q2n+1ab). (2.44)

Then we obtain Bn by substitution of x := c/(qa) in (2.43), in view of (2.35).From (2.38) one sees that Pn(x; a, b, c, d; q) is homogeneous of degree 0 in the three

variables x, c, d. Therefore, only three of the four parameters a, b, c, d are essential. Prob-ably for this reason, Gasper & Rahman [15, (7.3.10)] use the following notation for bigq-Jacobi polynomials:

PGRn (x; a, b, c; q) = Pn(x; a, b, c; q) := 3φ2

[q−n, abqn+1, x

aq, cq; q, q

].

Their notation is related to the notation (2.38) in the present paper by

Pn(x; a, b, c, d; q) = PGRn (ac−1qx; a, b,−ac−1d; q),

PGRn (x; q, b, c; q) = Pn(x; a, b, aq,−cq; q).

2.4. Little q-Jacobi polynomials. These polynomials are the most straightforwardq-analogues of the Jacobi polynomials. They were first observed by Hahn [16] and studiedin more detail by Andrews & Askey [3]. Here we will study them as a special case of thebig q-Jacobi polynomials.

When we specialize the big q-Jacobi polynomials (2.36) to c = 1, d = 0 and normalizethem such that they take the value 1 at 0 then we obtain the little q-Jacobi polynomials

pn(x; a, b; q) :=Pn(x; b, a, 1, 0; q)

Pn(0; b, a, 1, 0; q)

=2φ1(q−n, qn+1ab; qa; q, qx) (2.45)

=(−qb)−n q−n(n−1)/2 (qb; q)n(qa; q)n

3φ2

[q−n, qn+1ab, qbx

qb, 0; q, q

]. (2.46)

–32–

Here (2.45) follows by letting d→ 0 in (2.39), while (2.46) follows from (2.38) and (2.37).With the equality of (2.45) and (2.46) we have reobtained the transformation formula(1.38) for terminating 2φ1 series. From (2.46) we obtain

pn(q−1b−1; a, b; q) = (−qb)−n q−n(n−1)/2 (qb; q)n

(qa; q)n.

From (2.45) and (1.37) we obtain

pn(1; a, b; q) := (−a)n qn(n+1)/2 (qb; q)n(qa; q)n

.

Little q-Jacobi polynomials satisfy the orthogonality relations

1

Bq(α+ 1, β + 1)

∫ 1

0

pn(t; qα, qβ; q) pm(t; qα, qβ; q) tα

(qt; q)∞(qβ+1t; q)∞

dqt

= δn,mqn(α+1) (1− qα+β+1) (qβ+1; q)n (q; q)n(1− q2n+α+β+1) (qα+1; q)n (qα+β+1; q)n

. (2.47)

The orthogonality measure is the measure of the q-beta integral (1.27). For positivityand convergence we require that 0 < a < 1/q, b < 1/q (after we have replaced qα by aand qβ by b in (2.47)). It is maybe not immediately seen that (2.47) is a limit case of(2.42). However, we can establish this by observing the weak convergence as d ↓ 0 of thenormalized measure const.×w(x; qβ, qα, 1, d; q)dqx on [−d, 1] to the normalized measure in(2.47) on [0,1]:

limd↓0

∫ 1

−d(qβ+1t; q)nw(t; q

β, qα, 1, d; q) dqt∫ 1

−dw(t; qβ, qα, 1, d; q) dqt

= limd↓0

(qβ+1; q)n (−qβ+1d; q)n

(qα+β+2; q)n

=(qβ+1; q)n

(qα+β+2; q)n=

1

Bq(α+ 1, β + 1)

∫ 1

0

(qβ+1t; q)n(qt; q)∞

(qβ+1t; q)∞tα dqt.

Here the first equality follows from (2.41) and (2.22), while the last equality follows from(1.27).

Little q-Jacobi polynomials are the q-analogues of the Jacobi polynomials shifted tothe interval [0, 1]:

limq↑1

pn(x; qα, qβ ; q) =

P(α,β)n (1− 2x)

P(α,β)n (1)

.

If we fix b < 1 (possibly 0 or negative) and put a = qα then little q-Jacobi polynomialstend for q ↑ 1 to Laguerre polynomials:

limq↑1

pn

((1− q)x

1− b; qα, b; q

)=Lαn(x)

Lαn(0)

.

In particular, little q-Jacobi polynomials pn(x; a, 0; q), called Wall polynomials, are q-analogues of Laguerre polynomials.

–33–

Analogous to the quadratic tansformations for Jacobi polynomials (cf. [14, 10.9(4),(21) and (22)]) we can find quadratic transformations between little and big q-Jacobipolynomials:

P2n(x; a, a, 1, 1; q) =pn(x

2; q−1, a2; q2)

pn((qa)−2; q−1, a2; q2), (2.48)

P2n+1(x; a, a, 1, 1; q) =x pn(x

2; q, a2; q2)

(qa)−1 pn((qa)−2; q, a2; q2). (2.49)

The proof is also analogous: by use of the orthogonality properties and normalization ofthe polynomials.

Just as Jacobi polynomials tend to Bessel functions by

P(α,β)nN (1− x2/(2N2))

P(α,β)nN (1)

= 2F1

(−nN , nN + α+ β + 1;α+ 1;

x2

4N2

)

N→∞−→ 0F1(−;α+ 1;−(λx/2)2) = (λx/2)−α Γ(α+ 1) Jα(λx), nN/N → λ as N →∞,

little q-Jacobi polynomials tend to Jackson’s third q-Bessel function (1.60):

limN→∞

pN−n(qNx; a, b; q) = lim

N→∞2φ1(q

−N+n, abqN−n+1; aq; q, qN+1x)

= 1φ1(0; aq; q, qn+1x),

cf. Koornwinder & Swarttouw [24, Prop. A.1].

2.5. Hahn’s classification. Hahn [16] classified all families of orthogonal polynomialspn (n = 0, 1, . . . , N or n = 0, 1, . . .) which are eigenfunctions of a second order q-differenceequation, i.e.,

A(x) (D2q pn)(q

−1x) +B(x) (Dq pn)(q−1x) = λn pn(x) (2.50)

or equivalently

a(x) pn(q−1x) + b(x) pn(x) + c(x) pn(qx) = λn pn(x),

where A,B and a, b, c are fixed polynomials. Necessarily, A is of degree ≤ 2 and B isof degree ≤ 1. The eigenvalues λn will be completely determined by A and B. Onedistinguishes cases depending on the degrees of A and B and the situation of the zerosof A. For each case one finds a family of q-hypergeometric polynomials satisfying (2.50)which, moreover, satisfies an explicit three term recurrence relation

x pn(x) = pn+1(x) +Bn pn(x) + Cn pn−1(x).

(Here we assumed the pn to be monic.) Then we will have orthogonal polynomials withrespect to a positive orthogonality measure iff Cn > 0 for all n. A next problem is tofind the explicit orthogonality measure. For a given family of q-hypergeometric polyno-mials depending on parameters the type of this measure may vary with the parameters.

–34–

Finally the limit transitions between the various families of orthogonal polynomials can beexamined.

In essence this program has been worked out by Hahn [16], but his paper is somewhatsketchy in details. Unfortunately, there is no later publication, where the details have allbeen filled in. In Table 2 we give a q-Hahn tableau: a q-analogue of that part of the Askeytableau (Table 1) which is dominated by the Hahn polynomials. In the rφs formulas inthe Table we have omitted the last but one parameter denoting the base except whenthis is different from q. The arrows denote limit transitions. The box for case 3a, whichoccurred in an earlier version, is now suppressed, because it does not correspond to a classof orthogonal polynomials. Below we will give a brief discussion of each case. We do notclaim completeness.

Several new phenomena occur here:1. Within one class of q-hypergeometric polynomials we may obtain, depending on thevalues of the parameters, either a q-analogue of the Jacobi-Laguerre-Hermite class or ofthe (discrete) Hahn-Krawtchouk-Meixner-Charlier class.2. q-Analogues of Jacobi, Laguerre and Hermite polynomials occur in a “little” version(corresponding to Jacobi polynomials on [0, c], Laguerre polynomials on [0,∞) and Hermitepolynomials which are even or odd functions) and a “big” version (q-analogues of Jacobi,Laguerre and Hermite polynomials of arbitrarily shifted argument).3. There is a duality under the transformation q 7→ q−1, denoted in Table 2 by dashedlines. We insist on the convention 0 < q < 1, but we can rewrite q−1-hypergeometricorthogonal polynomials as q-hypergeometric orthogonal polynomials and thus sometimesobtain another family.4. Cases may occur where the orthogonality measure is not uniquely determined.

–35–

We now briefly discuss each case occuring in Table 2.

1) Big q-Jacobi polynomials Pn(x; a, b, c, d; q), defined by (2.36), form the generic case inthis classification. A(x) and B(x) in (2.50) are given by (2.34) and λn is as in the righthand side of (2.32). From the explicit expression for Cn in (2.44) we get the values ofa, b, c, d for which there is a positive orthogonality measure. For c, d > 0 these are givenby (2.23).

The q-Hahn polynomials can be obtained as special big q-Jacobi polynomials with−qad/c = q−N , n = 0, 1, . . . , N (N ∈ Z+). For convenience we may take c = qa. Theq-Hahn polynomials are usually (cf. [15, (7.2.21)]) notated as

Qn(x; a, b.N ; q) := 3φ2

[q−n, abqn+1, x

aq, q−N; q, q

]=

n∑

k=0

(q−n; q)k (abqn+1; q)k (x; q)k

(aq; q)k (q−N ; q)k (q; q)kqk.

Here the convention regarding lower parameters q−N (N ∈ Z+) is similar to the conventionfor (2.13).

q-Hahn polynomials satisfy orthogonality relations

N∑

x=0

(QnQm)(q−x; a, b, N ; q)(aq; q)x (bq; q)N−x

(q; q)x (q; q)N−x(aq)−x = 0, n 6= m.

The weights are positive in one of the three following cases: (i) b < q−1 and 0 < a < q−1;(ii) a, b > q−N ; (iii) a < 0 and b > q−N . For q ↑ 1 the polynomials tend to ordinary Hahnpolynomials (2.12):

limq↑1

Qn(q−x; qα, qβ, N ; q) = Qn(x;α, β,N).

2a) Little q-Jacobi polynomials pn(x; a, b; q), given by (2.45), (2.46), were discussed in §2.4.When we put b := q−N−1 in (2.46), we obtain q-analogues of Krawtchouk polynomials(2.17): the q-Krawtchouk polynomials

Kn(x; b, N ; q) := 3φ2

[q−n,−b−1qn, x

0, q−N; q, q

]= lim

a→0Qn(x; a,−(qba)

−1, N ; q)

with orthogonality relations

N∑

x=0

(KnKm)(q−x; b, N ; q)(q−N ; q)x(q; q)x

(−b)x = 0, n 6= m, n,m = 0, 1, . . . , N,

and limit transition

limq↑1

Kn(q−x; b, N ; q) = Kn(x; b/(b+ 1), N).

See [15, Exercise 7.8(i)] and the reference given there. Note that the q-Krawtchouk poly-nomials Kn(q

−x; b, N ; q) are not self-dual under the interchange of x and n.

–36–

2b) Big q-Laguerre polynomials, these are big q-Jacobi polynomials with b = 0:

Pn(x; a, 0, c, d; q) = 3φ2

[q−n, 0, qax/c

qa,−qad/c; q, q

](2.51)

=1

(−q−nca−1d−1; q)n2φ1

[q−n, c/x

qa; q,−qx/d

].

Here the second equality follows by (1.39). Note that

limq↑1

Pn(x; qα, 0, c, (1− q)−1; q) = 1F1(−n;α+ 1; c− x) = const. Lα

n(c− x).

Another family of q-analogues of Krawtchouk polynomials, called affine q-Krawtchouk

polynomials, can be obtained from (2.51) by putting −qad/c = q−N :

KAffn (x; a,N ; q) := 3φ2

[q−n, 0, x

aq, q−N; q, q

]= Qn(x; a, 0, N ; q)

with orthogonality relations

N∑

x=0

(KAffn KAff

m )(q−x; a,N ; q)(aq; q)x (aq)

−x

(q; q)x (q; q)N−x= 0, n 6= m, n,m = 0, 1, . . . , N,

and limit transition

limq↑1

KAffn (q−x; a,N ; q) = Kn(x; 1− a,N).

See [15, Exercise 7.11] and the references given there.

2c) q-Meixner polynomials

Mn(x; a, c; q) := 2φ1

[q−n, x

qa; q,−qn+1

c

]. (2.52)

For certain values of the parameters these can be considered as q-analogues of the Meixnerpolynomials (2.18), cf. [15, Exercise 7.12]. However, when we write these polynomials as

Mn(qax; a,−a−1; q) = 2φ1

[q−n, qax

qa; q, qn+1a

]= lim

d→∞Pn(x; a,−ad, 1, d; q)

and a is complex but not real, then these polynomials also become orthogonal (not doc-umented in the literature) and they can be considered as q-analogues of Laguerre poly-nomials of shifted argument. Moreover, these polynomials can be obtained from the bigq-Laguerre polynomials (2.51) by the transformation q 7→ q−1. Therefore, we call thesepolynomials also big q−1-Laguerre polynomials.

For a := q−N−1 the polynomials (2.52) become yet another family of q-analoguesof the Krawtchouk polynomials, which we will call affine q−1-Krawtchouk polynomials,

–37–

because they are obtained from affine q-Krawtchouk polynomials by changing q into q−1.These polynomials, written as

Mn(x; q−N−1,−b−1; q) := 2φ1(q

−n, x; q−N ; q, bqn+1) = lima→∞

Qn(x; a, b, N ; q),

where n = 0, 1, . . . , N , have orthogonality relations

N∑

x=0

(MnMm)(q−x; q−N−1,−b−1; q)(bq; q)N−x (−1)

N−x qx(x−1)/2

(q; q)x (q; q)N−x= 0, n 6= m,

and limit transition

limq↑1

Mn(q−x; q−N−1,−b−1; q) = Kn(x; b

−1, N).

See Koornwinder [21].

3b) Wall polynomials (cf. Chihara [12, §VI.11]) are special little q-Jacobi polynomials

pn(x; a, 0; q) = 2φ1(q−n, 0; qa; q, qx)

=1

(qna−1; q)n2φ0(q

−n, x−1;−; q, x/a).

The second equality is a limit case of (1.39). Wall polynomials are q-analogues of Laguerrepolynomials on [0,∞), so they might be called little q-Laguerre polynomials.

3c) Moak’s [29] q-Laguerre polynomials (notation as in [15, Exercise 7.43]) are given by

Lαn(x; q) :=

(qα+1; q)n(q; q)n

1φ1(q−n; qα+1; q,−xqn+α+1)

=1

(q; q)n2φ1(q

−n,−x; 0, q, qn+α+1).

The second equality is a limit case of (1.38). They can be obtained from the Wall poly-nomials by replacing q by q−1. Their orthogonality measure is not unique. For instance,there is a continuous orthogonality measure

∫ ∞

0

Lαm(x; q)Lα

n(x; q)xα dx

(−(1− q)x; q)∞= δm,n, m 6= n,

but also discrete orthogonality measures

∫ ∞

0

Lαm(cx; q)Lα

n(cx; q)xα dqx

(−c(1− q)x; q)∞= δm,n, m 6= n, c > 0.

Chihara [12]Ch. VI, §2 calls these polynomials generalized Stieltjes-Wigert polynomialsand he uses another notation. For certain parameter values these polynomials may beconsidered as q-analogues of the Charlier polynomials, see [15, Exercise 7.13].

–38–

3d) Al-Salam-Carlitz I polynomials

U (a)n (x) = U (a)

n (x; q) := (−1)n qn(n−1)/2 an 2φ1(q−n, x−1; 0; q, qx/a)

(cf. Al-Salam & Carlitz [1]) satisfy the three term recurrence relation

xU (a)n (x) = U

(a)n+1(x) + (1 + a) qn U (a)

n (x)− a qn−1 (1− qn)U(a)n−1(x).

Thus they are orthogonal polynomials for a < 0. They can be expressed in terms of bigq-Jacobi polynomials by

U (a)n (x) = Pn(x; 0, 0, 1,−a; q),

so they are orthogonal with respect to the measure (qx, qx/a; q)∞dqx on [a, 1], cf. (2.22).By the above recurrence relation these polynomials are q-analogues of Hermite polynomialsof shifted argument, so they may be considered as “big” q-Hermite polynomials.

3e) Al-Salam-Carlitz II polynomials

V (a)n (x) = V (a)

n (x; q) := U (a)n (x; q−1) = (−1)n q−n(n−1)/2 an 2φ0(q

−n, x;−; qna−1)

(cf. [1]). For a > 0 these form another family of q-analogues of the Charlier polynomials.

On the other hand, the polynomials x 7→ V(α/α)n (−qαx) can be considered as q-analogues

of Hermite polynomials of shifted argument. See Groenevelt [34] for a further discussionof this case.

4a) Stieltjes-Wigert polynomials

Sn(x; q) := (−1)n q−n(2n+1)/21φ1(q

−n; 0; q,−qn+3/2x)

(cf. Chihara [12, §VI.2]) do not have a unique orthogonality measure. It was alreadynoted by Stieltjes that the corresponding moments are an example of a non-determinate(Stieltjes) moment problem. After suitable scaling these polynomials tend to Hermitepolynomials as q ↑ 1.

4b) Al-Salam-Carlitz I polynomials U(a)n with a := −1 (these are also known as discrete

q-Hermite I polynomials hn( . ; q)):

hn(x; q) = U (−1)n (x; q) = qn(n−1)/2

2φ1(q−n, x−1; 0; q,−qx)

= Pn(x; 0, 0, 1, 1; q)

= xn 2φ0(q−n, q−n+1;−; q2, q2n−1x−2)

(cf. [1]), q-analogues of Hermite polynomials. By the last equality there is a quadratictransformation between these polynomials and certain Wall polynomials. This is a limitcase of the quadratic transformations (2.48) and (2.49).

4c) Al-Salam-Carlitz II polynomials V(a)n of imaginary argument and with a := −1 (also

known as discrete q-Hermite II polynomials hn( . ; q)):

hn(x; q) = i−n V (−1)n (ix; q) = i−n q−n(n−1)/2

2φ0(q−n, ix;−;−qn)

= limc→∞

Pn(x; ic, ic, qc, qc; q)

= xn 2φ1(q−n, q−n+1; 0; q2,−x−2)

(cf. [1]), also q-analogues of Hermite polynomials. By the last equality there is a quadratictransformation between these polynomials and certain of Moak’s q-Laguerre polynomials.This is a limit case of (2.48) and (2.49).

–39–

2.6. The Askey-Wilson integral. We remarked earlier that, whenever some nontrivialevaluation of an integral can be given, the orthogonal polynomials having the integrand asweight function may be worthwhile to study. In this subsection we will give an evaluationof the integral which corresponds to the Askey-Wilson polynomials. After the original eval-uation in Askey & Wilson [8] several easier approaches were given, cf. Gasper & Rahman[15, Ch. 6] and the references given there. Our proof below borrowed ideas from Kalnins& Miller [18] and Miller [27] but is still different from theirs. Fix 0 < q < 1. Let

wa,b,c,d(z) :=(z2, z−2; q)∞

(az, az−1, bz, bz−1, cz, cz−1, dz, dz−1; q)∞. (2.53)

We want to evaluate the integral

Ia,b,c,d :=1

2πi

∮

|z|=1

wa,b,c,d(z)dz

z, |a|, |b|, |c|, |d| < 1. (2.54)

Note that Ia,b,c,d is analytic in the four complex variables a, b, c, d when these are boundedin absolute value by 1. It is also symmetric in a, b, c, d.

Lemma 2.3

Ia,b,c,d =1− abcd

(1− ab) (1− ac) (1− ad)Iqa,b,c,d. (2.55)

Proof The integral ∮wq1/2a,q1/2b,q1/2c,q1/2d(z)

z − z−1

dz

z

equals on the one hand

∮wq1/2a,q1/2b,q1/2c,q1/2d(q

1/2z)

q1/2z − q−1/2z−1

dz

z

=

∮(1− az) (1− bz) (1− cz) (1− dz)wa,b,c,d(z)

q1/2 z (1− z2)

dz

z

and on the other hand

∮wq1/2a,q1/2b,q1/2c,q1/2d(q

−1/2z)

q−1/2z − q1/2z−1

dz

z

= −

∮(1− a/z)(1− b/z)(1− c/z)(1− d/z)wa,b,c,d(z)

q1/2 z−1 (1− z−2)

dz

z.

Subtraction yields

0 = q−1/2a−1

∮ {−(1− abcd)(1− az)(1− az−1) + (1− ab)(1− ac)(1− ad)

}

×wa,b,c,d(z)dz

z= 2πiq−1/2a−1

{−(1− abcd)Iqa,b,c,d + (1− ab)(1− ac)(1− ad)Ia,b,c,d

}.

–40–

By iteration of (2.55) and use of the analyticity and symmetry of Ia,b,c,d we obtain

Ia,b,c,d =(abcd; q)∞

(ab, ac, ad, bc, bd, cd; q)∞I0,0,0,0 . (2.56)

We might evaluate I0,0,0,0 by the Jacobi triple product identity (1.50), but it is easier toobserve that I1,q1/2,−1,−q1/2 = 1. (Note that this case can be continuously reached fromthe domain of definition of Ia,b,c,d in (2.54).) Hence (2.56) yields that I0,0,0,0 = 2/(q; q)∞.Thus

Ia,b,c,d =2(abcd; q)∞

(ab, ac, ad, bc, bd, cd, q; q)∞. (2.57)

By the symmetry of wa,b,c,d(z) under z 7→ z−1 we can rewrite (2.54), (2.57) as

1

2π

∫ π

0

wa,b,c,d(eiθ) dθ =

(abcd; q)∞(ab, ac, ad, bc, bd, cd, q; q)∞

, |a|, |b|, |c|, |d|< 1. (2.58)

Here w is still defined by (2.53). The integral (2.58) is known as the Askey-Wilson integral.

2.7. Askey-Wilson polynomials. We now look for an orthogonal system correspond-ing to the weight function in the Askey-Wilson integral. First observe that

1

2πi

∮(az, az−1; q)k(ac, ad; q)k

(bz, bz−1; q)l(bc, bd; q)l

wa,b,c,d(z)dz

z=

Iqka,qlb,c,d

(ac, ad; q)k (bc, bd; q)l

=(ab; q)k+l

(abcd; q)k+lIa,b,c,d.

Thus

1

Ia,b,c,d

1

2πi

∮ { n∑

k=0

(q−n, qn−1abcd; q)k qk

(ab, q; q)k

(az, az−1; q)k(ac, ad; q)k

}(bz, bz−1; q)l(bc, bd; q)l

wa,b,c,d(z)dz

z

=n∑

k=0

(q−n, qn−1abcd; q)k qk

(ab, q; q)k

(ab; q)k+l

(abcd; q)k+l

=(ab; q)l(abcd; q)l

3φ2

[q−n, qn−1abcd, qlab

ab, qlabcd; q, q

]

=(ab; q)l(abcd; q)l

(q−n+1c−1d−1, q−l; q)n(ab, q−n−l+1/(abcd); q)n

= 0, l = 0, 1, . . . , n− 1. (2.59)

Here we used the q-Saalschutz formula (1.61).The above orthogonality suggests to define Askey-Wilson polynomials (Askey & Wil-

son [8])

pn(cos θ; a, b, c, d | q)

a−n (ab, ac, ad; q)n=rn(cos θ; a, b, c, d | q) (2.60)

:=4φ3

[q−n, qn−1abcd, aeiθ, ae−iθ

ab, ac, ad; q, q

](2.61)

=n∑

k=0

(q−n, qn−1abcd; q)k qk

(ab, q; q)k

(aeiθ, ae−iθ; q)k(ac, ad; q)k

.

–41–

Since

(aeiθ, ae−iθ; q)k =

k−1∏

j=0

(1− 2aqj cos θ + a2q2j),

formula (2.61) defines a polynomial of degree n in cos θ. It follows from (2.59) that the func-tions θ 7→ pn(cos θ; a, b, c, d | q) are orthogonal with respect to the measure wa,b,c,d(e

iθ) dθon [0, π], so we are really dealing with orthogonal polynomials. Now

pn(x; a, b, c, d | q) = kn xn + terms of lower degree, with kn := 2n (qn−1abcd; q)n, (2.62)

so the coefficient of xn is symmetric in a, b, c, d. Since the weight function is also symmetricin a, b, c, d, the Askey-Wilson polynomial will be itself symmetric in a, b, c, d.

Proposition 2.4 Let |a|, |b|, |c|, |d|< 1. Then

1

2π

∫ π

0

pn(cos θ) pm(cos θ)w(cos θ) dθ = δm,n hn ,

where

pn(cos θ) = pn(cos θ; a, b, c, d | q),

w(cos θ) =(e2iθ, e−2iθ; q)∞

(aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ; q)∞,

hnh0

=(1− qn−1abcd) (q, ab, ac, ad, bc, bd, cd; q)n

(1− q2n−1abcd) (abcd; q)n,

and

h0 =(abcd; q)∞

(q, ab, ac, ad, bc, bd, cd; q)∞.

The orthogonality measure is positive if a, b, c, d are real, or if complex, appear in conjugatepairs,

Proof Apply (2.59) to

1

Ia,b,c,d

1

2πi

∮(pnpm)

((z + z−1)/2; a, b, c, d | q

)wa,b,c,d(z)

dz

z,

where pn is expanded according to (2.61) and pm similarly, but with a and b interchanged.

Note that we can evaluate the Askey-Wilson polynomial pn((z + z−1)/2) for z = a(and by symmetry also for z = b, c, d):

pn((a+ a−1)/2; a, b, c, d | q) = a−n (ab, ac, ad; q)n. (2.63)

When we write the three term recurrence relation for the Askey-Wilson polynomial as

2x pn(x) = An pn+1(x) +Bn pn(x) + Cn pn−1(x), (2.64)

then the coefficients An and Cn can be computed from

An =2knkn+1

, Cn =2kn−1

kn

hnhn−1

,

where kn and hn are given by (2.62) and Proposition 2.4, respectively. Then Bn can nextbe computed from (2.63) by substituting x := (a+ a−1)/2 in (2.64).

–42–

2.8. Various results. Here we collect without proof some further results about Askey-Wilson polynomials and their special cases and limit cases.

The q-ultraspherical polynomials

Cn(cos θ; β | q) :=(β2; q)n

βn/2(q; q)n4φ3

[q−n, qnβ2, β1/2eiθ, β1/2e−iθ

βq1/2,−βq1/2,−β; q, q

]

=const. pn(cos θ; β1/2, β1/2q1/2,−β1/2,−β1/2q1/2 | q)

=(β; q)n(q; q)n

n∑

k=0

(q−n, β; q)k(q1−nβ−1, q; q)k

(q/β)k ei(n−2k)θ

are special Askey-Wilson polynomials which were already known to Rogers (1894), howevernot as orthogonal polynomials. See Askey & Ismail [5].

By easy arguments using analytic continuation, contour deformation and taking ofresidues it can be seen that Askey-Wilson polynomials for more general values of theparameters than in Proposition 2.4 become orthogonal with respect to a measure whichcontains both a continuous and discrete part (Askey & Wilson [8])

Proposition 2.5 Assume a, b, c, d are real, or if complex, appear in conjugate pairs, andthat the pairwise products of a, b, c, d are not ≥ 1, then

1

2π

∫ π

0

pn(cos θ) pm(cos θ)w(cos θ) dθ +∑

k

pn(xk) pm(xk)wk = δm,n hn ,

where pn(cos θ), w(cos θ) and hn are as in Proposition 2.4, while the xk are the points(eqk + e−1q−k)/2 with e any of the parameters a, b, c or d whose absolute value is largerthan one, the sum is over the k ∈ Z+ with |eqk| > 1 and wk is wk(a; b, c, d) as definedby [8, (2.10)] when xk = (aqk + a−1q−k)/2. (Be aware that (1− aq2k)/(1− a) should bereplaced by (1− a2q2k)/(1− a2) in [8, (2.10)].)

Both the big and the little q-Jacobi polynomials can be obtained as limit cases ofAskey-Wilson polynomials. In order to formulate these limits, let rn be as in (2.60). Then

limλ→0

rn

(q1/2x

2λ(cd)1/2;λa(qd/c)1/2, λ−1(qc/d)1/2,−λ−1(qd/c)1/2,−λb(qc/d)1/2 | q

)

= Pn(x; a, b, c, d; q)

and

limλ→0

rn

(q1/2x

2λ2; q1/2λ2a, q1/2λ−2,−q1/2,−q1/2b | q

)=

(qb; q)n(q−na−1; q)n

pn(x; b, a; q).

See Koornwinder [23, §6]. As λ becomes smaller in these two limits, the number of masspoints in the orthogonality measure grows, while the support of the continuous part shrinks.In the limit we have only infinitely many mass points and no continuous mass left.

An important special class of Askey-Wilson polynomials are the Al-Salam-Chihara

polynomials pn(x; a, b, 0, 0 | q), cf. [6, Ch. 3]. Both these polynomials and the continuous

–43–

q-ultraspherical polynomials have the continuous q-Hermite polynomials pn(x; 0, 0, 0, 0 | q)(cf. Askey & Ismail [5]) as a limit case.

The Askey-Wilson polynomials are eigenfunctions of a kind of q-difference operator.Write Pn(e

iθ) for the expression in (2.61) and put

A(z) :=(1− az)(1− bz)(1− cz)(1− dz)

(1− z2)(1− qz2).

Then

A(z)Pn(qz) − (A(z) + A(z−1))Pn(z) + A(z−1)Pn(q−1z)

= −(1− q−n)(1− qn−1abcd)Pn(z).

The operator on the left hand side can be factorized as a product of two shift operators.See Askey & Wilson [8, §5].

An analytic continuation of the Askey-Wilson polynomials gives q-Racah polynomials

Rn(q−x + qx+1γδ) := 4φ3

[q−nqn+1αβ, q−x, qx+1γδ

αq, βδq, γq; q, q

],

where one of αq, βδq or γq is q−N for some N ∈ Z+ and n = 0, 1, . . . , N (see Askey &Wilson [7]). These satisfy an orthogonality of the form

N∑

x=0

Rn(µ(x))Rm(µ(x))w(x) = δm,n hn, m, n = 0, 1, . . . , N,

where µ(x) := q−x+qx+1γδ. They are eigenfunctions of a second order difference operatorin x.

There is nice characterization theorem of Leonard [26] for the q-Racah polynomials.Let N ∈ Z+ or N =∞. Let the polynomials pn (n ∈ Z+, n < N + 1) be orthogonal withrespect to weights on distinct points µk (k ∈ Z+, k < N + 1). Let the polynomials p∗n besimilarly orthogonal with respect to weights on distinct points µ∗

k. Suppose that the twosystems are dual in the sense that

pn(µk) = p∗k(µ∗n).

Then the pn are q-Racah polynomials or one of their limit cases.

Exercises to §2

2.1 Show that (at least formally) the generating function

e2xt−t2 =

∞∑

n=0

Hn(x)

n!tn

for Hermite polynomials (cf. [14, 10.13(19)]) follows from the generating function

(1− t)−α−1 e−tx/(1−t) =

∞∑

n=0

Lαn(x) t

n, |t| < 1,

for Laguerre polynomials (cf. [14, 10.12(17)]) by the limit transition

Hn(x) = (−1)n 2n/2 n! limα→∞

α−n/2 Lαn((2α)

1/2x+ α)

given in (2.7).

–44–

2.2 Show that (at least formally) the above generating function for Laguerre polynomialsfollows from the generating function for Jacobi polynomials

∞∑

n=0

P (α,β)n (x) tn = 2α+β R−1 (1− t+R)−α (1 + t+R)−β, |t| < 1,

where

R := (1− 2xt+ t2)1/2,

(cf. [14, 10.8 (29)]) by the limit transition

Lαn(x) = lim

β→∞P (α,β)n (1− 2x/β)

given in (2.5).

2.3 Show that (at least formally) the above generating function for Hermite polynomialsfollows from the specialization α = β of the above generating function for Jacobi polyno-mials by the limit transition

Hn(x) = 2n n! limα→∞

α−n/2 P (α,α)n (α−1/2x)

given in (2.6).

2.4 Prove that the Charlier polynomials

Cn(x; a) := 2F0(−n,−x;−;−a−1), a > 0,

satisfy the recurrence relation

xCn(x; a) = −aCn+1(x; a) + (n+ a)Cn(x; a)− nCn−1(x; a).

2.5 Prove thatlima→∞

(−(2a)1/2)n Cn((2a)1/2x+ a) = Hn(x)

by using the above recurrence relation for Charlier polynomials and the recurrence relation

Hn+1(x)− 2xHn(x) + 2nHn−1(x) = 0

for Hermite polynomials.

2.6 Prove the following generating function for Al-Salam-Chihara polynomials (notationas for Askey-Wilson polynomials in (2.60), (2.61)):

(zc, zd; q)∞(zeiθ, ze−iθ; q)∞

=∞∑

m=0

zmpm(cos θ; 0, 0, c, d | q)

(q; q)m, |z| < 1.

2.7 Use the above generating function in order to derive the following transformation for-mula from little q-Jacobi polynomials (notation and definition by (2.45)) to Askey-Wilsonpolynomials by means of a summation kernel involving Al-Salam-Chihara polynomials:

an pn(cos θ; a, b, c, d | q)

(ab, ac, ad; q)n

(ac, ad; q)∞(aeiθ, ae−iθ; q)∞

=

∞∑

m=0

pn(qm; q−1ab, q−1cd; q) am

pm(cos θ; 0, 0, c, d | q)

(q; q)m.

–45–

References

[1] W. A. Al-Salam & L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30 (1965),47–61.

[2] G. E. Andrews, q-Series: their development and application in analysis, number theory,

combinatorics, physics, and computer algebra, Regional Conference Series in Math.66, Amer. Math. Soc., 1986.

[3] G. E. Andrews & R. Askey, Enumeration of partitions: The role of Eulerian series and

q-orthogonal polynomials, in Higher combinatorics, M. Aigner (ed.), Reidel, 1977, pp.3–26.

[4] G. E. Andrews & R. Askey, Classical orthogonal polynomials, in Polynomes orthogo-

naux et applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni & A. Ronveaux(eds.), Lecture Notes in Math. 1171, Springer-Verlag, 1985, pp. 36–62.

[5] R. Askey & M. E. H. Ismail, A generalization of ultraspherical polynomials, in Studies

in Pure Mathematics, P. Erdos (ed.), Birkhauser, 1983, 55–78.

[6] R. Askey & M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal

polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300.

[7] R. Askey & J. Wilson, A set of orthogonal polynomials that generalize the Racah

coefficients or 6-j symbols, SIAM J. Math. Anal. 10 (1979), 1008–1016.

[8] R. Askey & J. Wilson, Some basic hypergeometric orthogonal polynomials that gen-

eralize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319.

[9] N. M. Atakishiyev & S. K. Suslov, The Hahn and Meixner polynomials of an imaginary

argument and some of their applications, J. Phys. A 18 (1985), 1583–1596.

[10] N. M. Atakishiyev, M. Rahman & S. K. Suslov, On classical orthogonal polynomials,Constr. Approx. 11 (1995), 181–226.

[11] W. N. Bailey, Generalized hypergeometric series, Cambridge University Press, 1935;reprinted by Hafner Publishing Company, 1972.

[12] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, 1978;reprinted by Dover Publications, 2011.

[13] A. Erdelyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Higher transcendentalfunctions, Vol. 1, McGraw-Hill, 1953.

[14] A. Erdelyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Higher transcendentalfunctions, Vol. 2, McGraw-Hill, 1953.

[15] G. Gasper & M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematicsand its Applications 35, Cambridge University Press, 1990; second edition 2004.

[16] W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math.Nachr. 2 (1949), 4–34, 379.

[17] M. E. H. Ismail, A simple proof of Ramanujan’s 1ψ1 sum, Proc. Amer. Math. Soc. 63(1977), 185–186.

[18] E. G. Kalnins & W. Miller, Jr., Symmetry techniques for q-series: Askey-Wilson poly-

nomials, Rocky Mountain J. Math. 19 (1989), 223–230.

–46–

[19] R. Koekoek & R. F. Swarttouw, The Askey scheme of hypergeometric orthogonal

polynomials and its q-analogue, Report 94-05, Delft University of Technology, Facultyof Technical Mathematics and Informatics, 1994; an extended version appeared asReport 98-17, 1998. A further extension is in the book R. Koekoek, P. A. Lesky& R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues,Springer-Verlag, 2010.

[20] T. H. Koornwinder, Group theoretic interpretations of Askey’s scheme of hypergeo-

metric orthogonal polynomials, in Orthogonal polynomials and their applications, M.Alfaro, J. S. Dehesa, F. J. Marcellan, J. L. Rubio de Francia & J. Vinuesa (eds.),Lecture Notes in Math. 1329, Springer-Verlag, 1988, pp. 46–72.

[21] T. H. Koornwinder, Representations of the twisted SU(2) quantum group and some

q-hypergeometric orthogonal polynomials, Indag. Math. 51 (1989), 97–117.

[22] T. H. Koornwinder, Jacobi functions as limit cases of q-ultraspherical polynomials,J. Math. Anal. Appl. 148 (1990), 44–54.

[23] T. H. Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the

SU(2) quantum group, SIAM J. Math. Anal. 24 (1993), 795–813.

[24] T. H. Koornwinder & R. F. Swarttouw, On q-Analogues of the Fourier and Hankel

transforms, Trans. Amer. Math. Soc. 333 (1992), 445–461; corrected version appearedas arXiv:1208.2521v1 [math.CA], 2012.

[25] J. Labelle, Askey’s scheme of hypergeometric orthogonal polynomials, poster, Univer-site de Quebec a Montreal, 1990.

[26] D. A. Leonard, Orthogonal polynomials, duality and association schemes, SIAM J.Math. Anal. 13 (1982), 656–663.

[27] W. Miller, Jr., Symmetry techniques and orthogonality for q-Series, in q-Series and

partitions, D. Stanton (ed.), IMA Volumes in Math. and its Appl. 18, Springer-Verlag,1989, pp. 191–212.

[28] K. Mimachi, Connection problem in holonomic q-difference system associated with a

Jackson integral of Jordan-Pochhammer type, Nagoya Math. J. 116 (1989), 149–161.

[29] D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81(1981), 20–47.

[30] F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974.

[31] M. Rahman & A. Verma, Quadratic transformation formulas for basic hypergeometric

series, Trans. Amer. Math. Soc. 335 (1993), 277–302.

[32] E. C. Titchmarsh, The theory of functions, Oxford University Press, second ed., 1939.

[33] J. A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11(1980), 690–701.

[34] W. Groenevelt, Orthogonality relations for Al-Salam-Carlitz polynomials of type II,arXiv:1309.7569v1 [math.CA], 2013.